Occasionally I’m told I have acquired a chest key. Where are the chests

You change into your purchased finery and equip those fabulous weapons in the same place you bought them, i.e. any peddler. Once bought, you simply click equip on the item and you will be wearing it. This also works for the various found items as well.

You can also change into a selection of outfits and weapons in the basement of the Manor after the end of the game. For some reason, before then the basement will show the outfits but not allow you to equip them. You browse the outfits by approaching the case directly behind the training dummy and the weapons are located in the bottom of the L on the same level.

The sockets are on balconies up high above the main room where the Animus and your Assassin buddies hang out. If you look sometimes you can see Juno pacing on the balconies. They're essentially climbing puzzles. When you get up to the balcony you'll see something that looks a little like a control console, you'll get a quick cutscene showing you placing the power source. IIRC they're only accessible after you get the power sources and a door slides open.

If you want a more spoilerish answer, this guide is pretty good.