How to write resume and apply to jobs after being a failed startup founder?

First, congrats on getting into Y Combinator! That's a hugely powerful brand and you can leverage that for your entire career. One of my regrets is that I didn't leave Big Tech to start a startup earlier.

I'll answer your questions in a few parts:

I've heard larger companies and recruiters in general don't like former founders

Where did you hear this? I don't think that's true, and in fact, I'd say the opposite. Or at least, I'd argue that if a company doesn't want entrepreneurial candidates, they are probably not desirable places to work for. Certainly OpenAI and Meta (the 2 companies you mention) love to hire startup founders.

• A huge source of innovation for larger companies is acquiring startups, so they clearly value those skills enough to pay a premium.
• If you look through bookface, you'll find tons of posts from founders who want to hire other founders. Many of those companies are too small for you, but imagine a former founder who is now an eng manager at a FAANG+: I guarantee they'd be interested in your profile.

The leadership at top companies wants people with ideas who take initiative and ownership. Those are the core skills of a good founder!

The evaluation criteria is likely different for recruiters, who are simply pattern-matching (most recruiters suck). They may actually have a negative bias against you, but you don't want your initial contact with a company to be the recruiter anyway (see my answer below).

Should I only apply to entry-level engineering jobs, and am I even eligible for them?

You're definitely eligible for entry-level engineering jobs -- that's why they're entry-level!

Given your 2 years of experience, you should aim higher and apply for mid-level roles (more on this in another response).

How should I write my resume/describe my experience+title?

If you want to apply for SWE roles, you should do it. You're objectively a great candidate.

• First, you have a strong background with high-powered internships, including at Meta. Emphasize that.
• Mention that you are a co-founder. The key is to focus on transferable skills in your experience. In 2 years of your startup, you must have done some engineering work or thinking. Talk about that and why that work was interesting.

I highly recommend watching this: [Masterclass] How To Write A Stellar Tech Resume That Gets You More Job Opportunities.

what should the ask be in the email if I'm not supposed to be asking for a referral in the email?

Two suggestions:

• Ask a question about their job or their career journey
• Ask a specific question you want advice about ("choosing between A or B")

How long should I be waiting for these "indirect connection" referrals to come through?

Depends on the timeline for your job search, I wouldn't wait more than 2 weeks if you have urgency in the job search. Redundancy is good here, BTW. You should not rely on a single contact to get you into a company. If the company is large enough (FAANG), you probably have multiple people you can chat with at in parallel.

Would reaching out cold to YC alumni who formerly worked at those companies be effective? I assume no, but asking anyways in case I'm misunderstanding.

They probably wouldn't be effective for referrals, but it doesn't hurt. Instead of viewing networking as a binary option (connect or not), view it as a priority queue. Former employees should be lower on your priority list compared to current employees.

On the topic of entry-level vs mid-level, I would simply defer that decision (as I talked about here):

• In many companies, your level is decided based on your interview performance. For example, when I interviewed at Facebook (aka Meta) in 2017, they put me between E4 (Mid-Level) and E5 (Senior). Based on my interview performance, I got the E5 offer. Generally there's less competition for more senior roles, so I'd opt for mid-level. 
• Automated application portals are your enemy. The way to win here is to leverage your network to get a referral at your desired company. Once you get a human on the phone (a recruiter), you can get your foot in the door and talk about the leveling decision later. With YC you have an amazing network already -- use it. 

Finally, the answer to if you should customize your resume to each different kind of job:

• If you're applying to different job functions (software engineer, data engineer, data science, etc), yes
• If you're applying for one job function (software engineering in your case), no.

For different companies or company sizes, I wouldn't customize your resume unless you really care about one company: it's not worth the time.

Once you get the interview, your resume doesn't matter much. And ideally you are landing interviews at the companies you want through your network.

Thanks so much for the amazing and supportive replies Rahul! Sorry for responding so late; I wanted to read through Taro to make sure I wasn't re-asking the same questions and be more thoughtful.

As you mentioned, it sounds like the key to win is my network (friends -> friends of friends -> YC alumni). I've read a few of the other posts on Taro regarding networking, but I think this situation is slightly different so I have a few more questions:

1. If the main goal would be to get referrals through YC alumni (i.e. the EM at FAANG+ that you mentioned or a big YC startup), how should I approach those cold emails? Outside of establishing credibility and showing I'm reaching out to specifically them, what should the ask be in the email if I'm not supposed to be asking for a referral in the email?
2. How long should I be waiting for these "indirect connection" referrals to come through? In another post on Taro, Alex mentioned it would be bad to wait a few weeks for a referral but the job gets taken down before it comes through (especially since the referrals will likely be weak since they don't know me).
3. Same question as #1 but with friends of friends, if there's any difference?
4. Would reaching out cold to YC alumni who formerly worked at those companies be effective? I assume no, but asking anyways in case I'm misunderstanding.