From now on, before opening a chest strike it with your weapon first, there are more of those mimics.
Do you remember the creepy giant snake at firelink? You can trade items you don't want with souls, although he won't give you much for worthless items it's still better than nothing. The more valuable an object, the more souls you get, but of course, usually valuable objects are things you may want to keep and use. Still, you get your inventory/bottomless bag list less clogged and get some souls in return.
For the moment though, just press on and try to take down the boss, so you can have the Lordvessel and teleport at will.
No, when you get to NG+ you do so with the same character, you get to keep all your items, weapons, amors, etc with a few exceptions (things like quest items and keys are taken away and you must find them again), you keep your level, and so on.
Also, no point in starting a new character just for pyromancy, because pyromancies don't scale with any stat; they're basically free magic, anyone can use it efficiently, to make the pyromancies more powerful you just have to improve the glove with Laurentius (you freed him, he should be in firelink, just before the giant snake) and bam! Pyromancies do more damage/are more effective. So, if you want to try pyromancies, equip the flame, equip a pyromancy spell and you're set.
Oh, to equip a spell/pyromancy/miracle you first need to have at least have 10 attunement for one slot, 12 for two, and so on. It goes up to a max of 10 slots but to have 10 slots you need to level attunement to 50, I wouldn't recommend doing something like that even for a mage character. For 16 points in attunement you can have 4 slots and those are more than enough, if you ask me.