Speaking from Experience, the Partners interviewing you probably won't even know which of the interviewees did or did not accept checks. Kirsty issues them as you ask before or after you interview and it is very casual.

I certainly don't think they would hold it against you even if they knew. They're good people and don't expect you to do anything for free

It's not a trick question. You'll get reimbursed and it won't affect your credibility in any way to ask for what is offered to you.

YC (likely) wouldn't blink an eye at paying travel expenses for a team they're interested in. The cost is insignificant to the potential upside.

That's a non issue. At the end of the interview, you go out in the main room, and will go and see their finance person to ask her to reimburse you, no question asked (bring a copy of your tickets in case she asks to see them, but I don't remember she did). You will get the answer later, and decision making is independent.

I know several groups who got in YC and asked for reimbursement for their travel. YC wouldn't offer it if it biased them in any way. They are sort of the "gold standard" in incubators.

I don't think so. At least for me the plane ticket is way over $600 (and I'm willing to spend a lot more just to be interviewed by these people)