I have an H-1B petition awaiting decision. Can I apply for a green card through a relative while it is pending?

You can but you should consult an attorneys about the specifics before you do so.

It should not as long as you can show that there is a reason for living in separate states, like work.

Indian applying for US asylum at the same time applying for Canadian PR

You are eligible to apply for asylum within one year of arrival.

EAD renewal for asylum pending case?

You need to write to the Vermont Service Center and include copies of what was submitted before.

I-601 Waiver for immigrant status ?

You may you qualify for a waiver but you immigration history, the criminal conviction, has to be reviewed by a lawyer.

Can I file I-130 with I-485 for my son who is just 11 years old but his custody belongs to my ex wife who lives out if US?

Your child may derive citizenship from you if you meet the requirements of N-600.

Please see a lawyer with that letter.

How will it affect my status/chances to get back if I withdraw my O1B application after receiving RFE and leaving the US?

Yes you can and it might actually help you not to accrue unlawful presence.

Early filing?

You can file 90 days before your 5th anniversary as an LPR.

Can I apply for travel authorization (advance parole) if my country's TPS designation is ending soon?

Generally speaking yes, but its very case specific so consult an attorney please.