Since early age, have heard Ulema describing instance of how H. Ali a.s was named Hayder by his Honoured Mother.

1. Imam Ali was alone in the house and a snake came. Once his mother came back, she saw snake ripped into pieces. She uttered Hayya Darra which was then routinely called Haydar.
2. Arabs had custom to check legitimacy of a child through snake and same snake was allowed to sniff H. Ali so he ripped into 2 pieces and that custom was stopped.

Both of above instances are read under the pretext how he was named hayder

1. Done above narrations have any relevance?
2. How is it possible that H. Abu Talib a.s allow such thing to happen in his house?
3. What if Snake found the child illegitimate and swallow, do their parents allow this to happen in their presence?
4. What is the Real meaning of name Haydar?

This is the hadith you mentioned and it’s source.

عن أنس، عن عمر بن الخطاب: أن علياً "عليه السلام" رأى حية تقصده وهو في المهد، وقد شدت (وشدت) يداه في حال صغره.

فحول نفسه، وأخرج يده، فأخذ بيمينه عنقها، وغمزها غمزة[1]، حتى أدخل أصابعه فيها، و أمسكها حتى ماتت.

فلما رأت ذلك أمه نادت واستغاثت، فاجتمع الحشم، ثم قالت: كأنك حيدرة [حيدرة] اللبوة، إذا غضبت، من قبل أذى أولادها

مدينة المعاجز ج2 ص35 ومناقب آل أبي طالب ج2 ص287 و 288 و (ط المكتبة الحيدرية) ج2 ص120 وبحار الأنوار ج41 ص274 والإمام علي بن أبي طالب "عليه السلام" للهمداني ص611.

Haydara means a fierce lion

Just because the hadith is mentioned doesn’t really mean it is authentic. However, we know that Imam Ali was brave from a young age, and just as Allah chose certain prophets to show signs of greatness from a young age, that could very much be the case with Imam Ali.

The part of Arabs had a custom to check legitimate birth of child, I have never heard of.