Rear replacement housings are almost always a bad fit, the only ones I have ever managed to fit properly are the ones with the bezel already attached to it. As for the LCD try to make sure no dirt or debris is in/around the connectors on both the board and the LCD flex cable. It may be worth cleaning them both with isopropyl too (make sure to use 99% or above and a clean cotton bud or piece of microfiber cloth). If this doesn't work then the LCDs ribbon cable may be damaged and the whole part should be replaced.
First try with a different charger, if that doesn't work replace the dock connector and if that doesn't work order a new battery. I prefer this order as a new dock connector is easier than replacing a battery and trying a different charger/plug socket is the easiest of them all! It may also be a good idea to look inside the dock connector for fluff and bits of tissue from your pocket and clean these out as they can occasionally prevent charging
The best way to test this is to use the voice recorder app and make sure that the "speaker" button in the top left corner of the screen isn't enabled - this means that the sound recording will play through the ear speaker. If you can't hear anything then make sure that ribbon cable number 3 is fully pushed in, I like to use a pair of tweezers to help me and then the blue opening tool to flip down the clamp on the logic board. If you still can't hear anything then you may have a faulty part, if you bought from eBay they should sent you another without a problem - in the UK I pay £15 for the whole mid-frame, ear speaker and home button and I have never had any problems in this price range! I think that's around $20 As for the screen brightness you could try using another LCD to see if you get the same problem. If you do you might have a bent pin on the logic board which requires a difficult repair.