Why did actresses like Babita and Neetu Singh stop acting after marriage, and how has this trend shifted in modern Bollywood?

 Because of the patriarchal setup. According to the tradition of the Kapoor family, the women of the family are not supposed to work in films. So both Babita and Neetu Singh had to stop acting after their marriage.

Raj Kapoor also believed that filmy bahus don't make good traditional housewives, so he was very much against the marriage of his son to actresses. But Randhir Kapoor and Rishi Kapoor managed to marry with girls of their own choice. But Raj Kapoor put his foot down for the third son Rajeev Kapoor who was not allowed to pursue actresses Padmini Kolhapure and Divya Rana. Raj Kapoor even threatened Padmini to throw her out of Prem Rog if the romance continued. Rajiv Kapoor blamed his father for his personal and professional debacles. Shashi Kapoor’s daughter and wife also worked in films and he produced his wife's film 36 Chowringee Lane, but refused to launch his children. Shammi Kapoor's wife also left films after marriage.

This trend changed with Karishma Kapoor. Babita was a highly ambitious lady and she wanted to see her daughters successful. Also the career of Randhir Kapoor remained patchy due to which they needed financial stability. So Babita at the risk of tainting her marriage with Randhir groomed Karishma to become an actress, and she became the first daughter of Kapoor Khandan to work as an actress. This was followed by Kareena Kapoor and their success opened the gateway for women working in the films. Even Neetu Kapoor made her comeback in 2009 and since then has been working on limited projects. Now even the daughter of Neetu Kapoor, Riddhima has made her debut with her mother in Dadi ki Shadi. And Alia Bhatt is an established actress already.

This tradition was common in many Bollywood families wherein actresses left movies after marriage. Amitabh asked Jaya to stop working in films after marriage at the time when she was at the peak of her career. She returned occasionally to films but sacrificed her larger career. Dilip Kumar wanted Madhubala to leave movies after marriage and the relation couldn't sustain. Sunil Dutt was adamant that no women of her family should work in films. So Nargis Dutt left movies after marriage even though she was perhaps a bigger star than Sunil Dutt at that moment. She only returned to do one film for her brother and won a National award for that. And Sanjay Dutt continued this tradition of not allowing his wives (Richa Sharma and Manyata) to work in films and even when his daughter Trishala wanted to work in films he didn't let her.

What are the lists of some actresses which have simple personality in South Indian film industry?

 Keerthy Suresh(most simple one according to me)

Kajal Agarwal

Sai Pallavi

Rashmika Mandanna

Rakul Preet Singh

Anupama parameswaram

Sree Leela

Who are the talented but underrated actresses in Bollywood?

 There are quite many Talented yet Underrated Actresses in B'wood. But Some of the more prominent ones are:-

Konkana Sen Sharma

2. Tabu

3. Nandita Das

4. Tisca Chopra

5. Richa Chadda

6. Divya Dutta

7. Radhika Apte

8. Swara Bhaskar

9. Kirti Kulhari

10. Sayani Gupta

11. Kalki Koechlin

12. Shweta Tripathi

Edit:-

Tillotama Shome and Rasika Duggal for Qissa

Tanishtha Mukherjee

Surveen Chawla for Parched and Ugly

Edit 2:-

For those who're saying that 'These Actresses aren't Underrated because they've Won National Award'. This Argument is wrong! Just because they've Won National Award, doesn't means that they're Treated at par with other ‘Mainstream A Listed Actresses'. Underrated, here means that they don't get same attention from Media, Mainstream Award Shows and Big Production Houses. These Actresses usually don't get Same roles as Deepika Padukone or Kareena Kapoor.

Edit 3:-

Sorry for this late addition here but lately I remembered her, so thought to list her.

Nimrat Kaur (Honoury Mention)

Though some might think and start sayin that 'she's not underrrated' et cetera et cetera but I believe that she indeed is an Underrated Actress. After proving her Metal again and again, Even Hollywood gave her big projects and yet Our Nepotism filled industry (Bollywood) is still not Casting her in Women Centric and Strong Protagonist Roles.

Which Bollywood actress had a very short career span and then disappeared?

 A lot of names have already been mentioned in other answers. So, I will try to add a few more names which have been missed.

Gracy Singh : The graceful and beautiful actress from Lagaan, seems you have disappeared. She acted in few more movies but she is not seen making any presence on-screen since quite a few years.

Anita Raj : She did good nunber of movies and was popular too but she disappeared after those movies. She made a comeback on small screen.

Amrita Arora : Not that she had amazing acting skills or anything but she tried for few years but now she has almost disappeared.

Koena Mitra : TThere was a phase when her presence could be felt very much on-screen. But, gradually she disappeared. She had acted in movies like Apna Sapna Money Money.

Payal Rohtagi : She had made numerous controversial statements and I believe she still does that. However, she seems to have disappeared from the big screen.

Yukta Mukhee : The beautiful actress had started in few movies but she could not make it big.

Diana Hayden : Another beautiful lady who had no luck in Bollywood and disappeared from Bollywood.

Trisha Krishnan : She is a popular name in the south movie industry bit disappeared from Bollywood after doing Khatta Meetha.

Kajal Agarwal : Another popular name from south movie industry who appeared in the Hindi movie Singham and then disappeared.

Image sources : Pinterest

Who are some Bollywood actors/actresses who got married at a very young age?

 As of latest Update

Actress Alia Bhatt was 29 age when she married to Ranbir Kapoor

Alia Bhatt married actor Ranbir Kapoor on April 14, 2022

Interesting thing is Alia Bhatt and Ranbir Kapoor have an 11-year age gap. Ranbir is older, born on September 28, 1982, while Alia was born on March 15, 1993

By Bollywood standards today, 29 is relatively young, but not unusually youngest. Many older-generation stars married in their late teens or early 20s.

⭐List of popular Bollywood Celebrities who got married at very young age

1. Actress Bhagyashree (Born on February 23, 1969), She was 19 years old at the time of her wedding on January 19, 1989, married to Himalaya Dasani

2. Shashi Kapoor was 20 age when he married Jennifer Kendal in July 1958. He was born on March 18, 1938, and passed away on December 4, 2017, at the age of 20 years old

3. Actress Neetu Singh was 21 age when she married to Rishi Kapoor on (1980)

4. Babita was Around 23 when she married to Randhir Kapoor (1971)

5. Divya Bharti was 18 when she married to Sajid Nadiadwala (1992)

6. Actor Shah Rukh Khan was 25 years old (born November 2, 1965) and Gauri was 21 or 22 years old (born October 8, 1970) married on October 25, 1991.

7. Akshay Kumar and Twinkle Khanna got married on January 17, 2001. At the time of their wedding, Akshay Kumar was 33 years old (born September 9, 1967) and Twinkle Khanna was 27 years old (born December 29, 1973).

Interesting thing is Twinkle Khanna had Short Span time of Career in Movies, after marriage she retired from acting.

Twinkle Khanna acted in a total of 15 movies

(including one Telugu film) during her active acting career from 1995 to 2001, so around 6–7 years …before later transitioning into a successful author, columnist, and film producer

8. Saif Ali Khan was 21 years old and Amrita Singh was 33 years old (frequently cited as October or late October 1991). At the time of their wedding, October 1991 got married in Saif Ali Khan, marking as 12 year age gap difference

Later Saif Ali Khan Divorced and He married to Kareena Kapoor and Saif Ali Khan got married on October 16, 2012. At the time of their wedding, Saif Ali Khan was 42 years old and Kareena Kapoor was 32 years old, reflecting an age difference of about 10 years.

9. Late actor Dharmendra was 19 years old when he married his first wife, Prakash Kaur, in 1954, long before he started his acting career in Bollywood. He was approximately 45 years old when he married his second wife, Hema Malini, in 1980.

10. Dimple Kapadia was 16 years old when she married actor Rajesh Khanna on March 27, 1973. The wedding took place shortly before the release of her debut film, Bobby.

Dimple Kapadia and Rajesh Khanna never officially or legally divorced, although separated in 1982 after having two daughters (Twinkle and Rinke), and lived apart for decades, they remained legally married until Khanna’s death in 2012.

11. Amitabh Bachchan was 30 years old and Jaya Bachchan (then Jaya Bhaduri) was 25 years old when they got married on June 3, 1973

12. Actor Anil Kapoor was 27 years old and Sunita Kapoor (Costume Designer) was 19 years old when they got married on May 19, 1984.

13. Bobby Deol was 27 years old when he married Tanya Ahuja on May 30, 1996. Tanya was 19 years old at the time.

14. Sunny Deol was 27 or 28 years old when he married Pooja Deol in 1984. he married her shortly after his breakout Bollywood debut film Betaab (1983

15. Actress Kajol was 24 years old when she married fellow actor Ajay Devgn on February 24, 1999.

16. Sonali Bendre was 27 years old when she married filmmaker Goldie Behl on November 12, 2002. She was born on January 1, 1975.

17. Sanjay Dutt was 28 age .(first marriage with Actress Richa Sharma in 1987), second marriage in 1998), and in 48 age (third marriage in 2008, married to Maanayata Dutt)

He has been married three times. His first marriage wasn't divorce, rather Richa tragically passed away from a brain tumor in 1996.

18. Juhi Chawla was 27 to 28 years old when she married industrialist Jay Mehta in December 1995. She was born on November 13, 1967, and her husband was born on January 18, 1961, making him roughly 34 years old at the time of their wedding.

Among major Bollywood stars, Actress Dimple Kapadia (16) and Late Actor Dharmendra (19) are among the celebrities who got married at Young Age.