1) Yes, you can run your Python programs remotely. You can run them even when you log off in a tmux window or as a systemd service since it appears that VPS69 is running Ubuntu.
2) You can use FTP/SFTP to move files to your server. Alternatively, you can just use the scp command to move files.
3) There are a couple of ways to make flask content showup on a domain. You can just tell flask to run on port 80 or 443 (you'll need to run it with sudo if you do this). You could also use nginx or apache or manage your flask content. This might be harder to do depending on your familiarity/comfortability with the command line. Either way, you would need to also add an A record to your VPS where you control your domain's DNS.
4) You would have to install a remote desktop server on your VPS. I personally wouldn't recommend it for your use case though since in my eyes, it's just a waste of system resources.
There are plenty of guides online to do what you've asked for in your post. I'm sure there are also community members that can provide pointers, but ultimately you'll have to learn on your own since no one is going to hold your hand forever.