I don't think there's a way to include your Gmail signature that you set up in Gmail itself (in the website interface) but you can easily add text for a signature in the outgoing text itself. But, to be honest, I don't know; I've never researched this before.

I don't think so either. I did some searching this weekend and didn't find a way.

Signatures usually aren't synced across platforms.  The Gmail website, Gmail Android app, Gmail iOS app, Outlook etc. all use their own signatures.  If you want to share signatures, you have to copy and paste them manually.  Send yourself a message with the signature, then copy the signature onto the app you are using.

Yeah, that's right. I can't use my standard web-based Gmail signatures on my Android, but that's okay because I want a different signature on my phone anyways. That says, "Hey, I'm replying from my phone, so it's going to be short, to the point, with voice recognition, and don't expect the novel." LOL.