1.As my games were released on Steam, of course the source code contains the standard Steam verification (we can call it a DRM), Steam achievements etc. Do I have to remove them before releasing here? If not, I guess people have to start Steam too, am I right? Is it a problem?

I do not recommend uploading a version of your game that doesn’t run if the player doesn’t have Steam. Your page will be reported as broken.

2.Should I insert some itch.io-specific code into the source?

It’s not necessary, most content on our platform is DRM free

3.If I’d like to add Steam keys, where can I upload them?

You can add them on your project’s edit page, we call them external keys. You can learn more here: https://itch.io/docs/creators/download-keys#external-keys-steam-etc

4.Is the key-giving process automatic, or do I have to do something manually after a player buys a game?

It is automatic. Your only responsibility is to ensure that there are enough keys stocked for your buyers, and to add more if they are running low