1.) Am I supposed to trim away extraneous leaves from sweet pepper plants or not? My peppers are small and are suffering from some sort of bottom mush/mold issue.
Do you mean the bottom of the peppers, or the bottom of the plant? If it is the bottom of the peppers themselves, it could be blossom end rot. It normally affects tomatoes, but it also affects peppers occasionally, and eggplant, from what I have read. I would pick off the peppers and wait and see if does not correct itself by the next set of peppers. And the next time you plant peppers, work a fair bit of organic matter into the soil around them to try and prevent this.
I myself seldom trim away any leaves on peppers plants. I usually make it a practice to pick the first crop of peppers though, and pinch off the first set of blooms while the plant is still small, if for no other reason than to encourage the plant to grow somewhat larger. It seems that in my climate peppers usually try to start producing too early, and these peppers are usually small themselves.
2.) I've never grown strawberries before and I have two gorgeous containers full of happy plants. Now that they have stopped producing: what do I do with the plants??
If the container looks exceptionally crowded, you might want to thin them out a little this fall. Plants and runners that you thin from this container could go into another container, to increase your production for next year. Also, your plants will usually produce quite a bit more fruit in the second year than the first. They overwinter pretty well in the south if you give them a good layer of mulch or straw, and put the containers in a semi-sheltered location where the wind can't get to them this winter. Inside a garage or outbuilding works perfectly.