Well I do have CS:S, Half life 2 and Episode 2 installed and I do have everything from the workshop installed.
Then, on your main menu on Garry's Mod, or pause menu whilst on a map/server, there's a small controller icon in the bottom right of your screen, typically next to a flag denoting your language of choice. If you click on that, it will open a list of source games.
Locate the games I mentioned. There will either be a red X or a square, or a square with a tick. If the square does not have a tick, click that square to tick it.
This will mount the games, enabling Garry's Mod to be able to use the games. Then, restart your Garry's mod. Check if they are still ticked, if so, then you should be able to use the server.
If you encounter any other problems, go onto your Steam Library, find the games I mentioned, along with Garry's Mod, right click on them, check their properties, go to "Local Files" and click "Validate Files", this will scan your installed games for signs of errors, and will fix/redownload any missing files, which if you've followed my above, have them all via steam and followed this step, there should be nothing stopping you from playing the server.
In addition, validating the files should also solve stuff like your problem with anims, unless you have replacers via the workshop.