I do not know why you are not responding, but you are not charged per question, per response, nor per answer. So after I give you my answer, you can continue to ask me questions without additional charge until you are satisfied. I apologize if it is a site issue that you posted and the post did not go through.
Let's go one by one:
1- Would it be better that I claim for asylum on the merits of apostasy or coming out as gay recently?
You can include both reasons.
2- Would the combination of both, for example changing religion to Christianity as a result of turning gay help my case, or would it make the whole thing sound fishy?
Well, you have to be careful because some Christian denominations do not accept homosexuality. So it would look fishy if you changed to a denomination that did not accept homosexuality.
3- Can it decrease my chances of getting asylum the fact that I'm on H1B visa right now? (The officer thinking that I don't really need an asylum since I'm already working...)
No. Not at all.
3.5- Is it true that if I have H1B, my case wouldn't be passed to immigration judge if not accepted in the first step?
It is not true.
4- Once the first year of staying in the US lapses, does it matter whether such mentioned cases be brought up after 4 years of legally staying in the US or 6 years? (Considering that the case is brought in reasonable amount of time after it's known.)
Well, that will be problematic. Of course, you would have to prove the timing of your change of religion and also when you determined that you were gay. If they don't believe that it was all recent, then you would most likely be denied Asylum because of the 1 year filing deadline, but you could be granted Withholding of Removal which doesn't give you Residency, but at least it is protection against deportation.
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