The speaker is almost certainly the cause of the garbled/low volume/staticky sounds. Replace it with the correct speaker for a 5 volt PS2 engine. Just get the replacement speaker from MTH. These are out of stock at MTH and they don't really know when they are going to get them.
By the engine "not remembering its name", I expect that what you mean is that you have to keep re-adding it to the DCS Remote after you've turned off track power. No. It stays in the remote through a power cycle. It is only when you do a Feature Reset or Factory Reset that you lose the name. You would not normally do a reset, but it is strange behavior and somewhat worrisome. If so, it's most likely either a dead/weak battery or a disconnected wire between the battery and the PS2 board. Less likely is that the board's battery charging circuit has become defective.
Place the engine on a powered track and start it up making sounds. Turn off track power. If sounds don't persist for 6-7 seconds, first ensure that the wires have continuity between the battery and the board. If so, replace the battery with a new 8.4 volt NiCad or NMH. Using the 9volt alkaline battery temporarily, the sound continues for 7 seconds after turning off the power.
I hope I have responded to your suggestions.