Mentioning the layoff might not be avoidable because you're likely to get recruiters asking you why you're looking for a new job. That said, most recruiters and hiring managers understand that a lot of the layoffs that happened aren't necessarily performance related, so you can justify it by saying like "Oh my company decided to reprioritize and the project I was on got cut, so they let me go," and that shouldn't cause any issues for you. You should be fine not mentioning the baby until late in the process (like after you accept the offer lol). You're definitely not going to get any questions about that (if you do, it's a red flag, run away). If you plan to take paternity leave or something right away though, I would make sure to check if the company has any probation period on eligibility for it. Some companies won't let you use the full benefit until you've been there at least a year. And it's probably worth letting your hiring manager know earlier on that you anticipate doing that.

Super helpful, I really appreciate the input!