There's a display case which needs a key to be opened. What key is it? I found a rusty key in the mages guild but it doesn't fit it. It has these cool crystals inside of it, which I don't know what they are or how to get them.

Also, there's a chest in The Main Ingredient (shop in Imperial City) and there's a chest that requires a key, but the store key doesn't work for it so I was wondering what does.

The case with the crystals is part of a side quest, I don't think there is a key for it. The other chest, I don't know about. I've heard of people finding merchant chests in the game but I haven't found them. If it's a merchants chest you had best leave it alone, you might need that merchant later.

Is there a (non complicated) way of opening both of these chests? And could the crystals be from the Frostfire Glade sidequest? If so, you get no key from it. As for teh merchant chests, try looking in the shop and it's there.

The crystals there are for decorytive Puposes only. You could unlock the case via console. Open the console, then click on the Display Case and type "unlock". But then, you cant pick up the Crystals.