It's a total misconception, Laxman was not a hot headed and angry young man. Laxman's character is one of the most glorious ones in Ramayana.
Laxman and Shatrughna were sons of Dashrath and Sumitra. Laxman followed Shri Ram and Shatrughna followed Bharat.
Laxman was a very intelligent, wise and very good warrior who defeated Indrajit, not an easy achievement at all. He is called Avatar of Sheshnaga in Valmiki Ramayana.
Unlike what TV serials have shown us, Laxman did not rebuke Janaka for insulting Kshtriyas in Swayamvar because there wasn't any Swayamvar at all. He never spoke before elders unnecessarily, specially when Shri Ram was present.
Laxman also didn't converse with Parshuram at all. It's Shri Ram who did all the conversation with Parshuram.
The one time he was extremely angry was when Kaikeyi demanded his exile and Dashrath could'nt stop it. He advised Shri Ram to imprison Dashrath and if necessary even kill him as he failed to perform his duties as King and agreed to send Shri Ram in exile without any fault. Shri Ram reasoned with him and then only his anger toned down. He still wasn't happy Shri Ram wasn't fighting for his rights and called him too much believer in fate.
Laxman was a highly devoted and intelligent brother. He's the one who said that Golden deer was the demon and it was a trick. Shri Ram said if it's a demon he would kill him and if it isn't, he would bring him as Sita wanted the deer.
Laxman saw Sita like a mother, he never directly looked at her face and always saw her feet because his eyes were always low. That's why when Sita accused him saying unworthy words, Laxman was distraught and left Sita to find Shri Ram.
There is no Laxman Rekha in Valmiki Ramayana. He left Sita because she was continuously saying words which pinched him and he prayed to jungle gods to keep her safe. Lord Ram later scolded Laxman to not have left Sita because she said those words out of anger, fear and distress and she didn't actually mean it.
When Sita was abducted, Laxman is the one who gave courage to Shri Ram and motivation to find her. He has done it repeatedly in the epic, many times he's the sole moral and emotional support to Shri Ram.
In short Laxman was not short tempered, he had warrior like qualities but it doesn't mean he was roaming around brooding and looking angry all the time.