It won't lemme connect my Steam to my GOG in "integrations". It keeps failing. It says "offline", then logs me back out. I've logged into my Steam dozens of times now

Also, can I play my GOG games on Steam? If so, do I unlock achievements on Steam from my GOG games? Are the libraries integrated? One in the same? Or can I only play my Steam games in GOG?

In some other planar universe, this may work. But you are trying marry Burger King and McDonalds and ask why the relationship is sour sauce, on a rancid green bun!

Ignore the unhelpful troll above, go ask questions here - https://www.gog.com/forum/general_beta_gog_galaxy_2.0

Solve your Steam integration here - https://www.reddit.com/r/gog/wiki/galaxy-integrations

For your other questions:

1. You can play GOG games on Steam. Install it and then manually add the game to Steam as shown here - https://help.steampowered.com/en/faqs/view/4B8B-9697-2338-40EC

2. You can't unlock achievements on Steam with your GOG games unless in rare cases where the dev coded for both to work that way like the Nihon Falcom games published by XSeed/Marvelous. If you have Steam running with your GOG game on and try to activate an easy achievement and it doesn't, chances are it's not achievement cross-compatible.

3. The libraries are separate, but can be grouped together through Steam customizable to your order and categories/groups.

4. In general, you'll still need the Steam launcher to play Steam games. Galaxy is just a all-in-one-hub that opens Steam for you when you want to play a Steam game with one click.

If you care more about Steam achievements, I'd say it's in your better interest to just stick to buying from Steam only.

The question is already answered in GOG Galaxy Community Integrations – Here's how to fix them! (Steam, EA etc..)