Is there a downside of having a Poke-stop at your work?

There is luckily enough a Pokestop close enough to my workplace to check-in every few minutes. This yields me the necessary Pokeballs and experience, but my bag is getting full. I am only allowed to have 350 items. Are there rare items (so no Pokeballs) that I would miss out on with a full bag? Is it a good habit to not check-in very often to keep an empty bag?

Just keep using the Pokestop. If your bag is getting full you can bin a few pokeballs, don't worry about binning them all since it asks you which amount you want to throw away.

This way, you can keep getting exp and chances for the rarer items which you do want to keep. Good luck!

As you get higher level you get more different items(at level 5 you start getting potions and revives, I don't know if you get more different things at higher levels). But if you keep catching Pokemon you will use your pokeballs, so you can also use an incense to get more Pokemon in the area. You can also place a lure module which attracts Pokemon to the PokéStop.

So no, keep using the Pokestop, you'll get free xp as well

Please note that you don't need to have any space left to get the 50xp bonus.

So while going up in levels (and thus item types) you can keep an acceptable amount for each one and still get the xp without having to worry about having any empty space in your bag.