Can I use Time Machine on my MacBook Air and back up files from my old MacBook Pro that I no longer have?

You can restore files backed up by the Mac you no longer have:

Restore items backed up with Time Machine on Mac - Apple Support

Yes; your question sort of confused me. If you deleted something prior to creating an initial Time Machine backup, it wouldn't have been backed up.

However, if all you did was drag it to the Trash and did not empty it, then it will have been backed up. However, you can't just open the Trash folder and "Enter Time Machine" because its interface will just default to your Home folder.

There is a way to open the Trash folder and use Time Machine to restore its contents, but it involves a bit of work. If you think that's a possibility please write back. I would need to try it before providing instructions, and I'd rather not do that 😆

No; I understand what you want to do. But if that's the earliest date TM recorded, then nothing precedes it.

If it's any consolation I'm trying and failing to find a way to get Time Machine to open backed up Trash folders. Perhaps it can't be done any more. I'll set it aside for now.