no issues with my DE LW3 (quietest pin I own in fact), friend has a TFTC with no hum. I would say a part missing, like a ferrite bead (as suggested), or a loose ground somewhere.

Doubting bad speaker, usually when a speaker goes bad though it still functions without noise (might crackle when you adjust the sound pot). My space shuttle did this, replaced the bottom speaker with a new $16 pyle, crackling went away (not surprising for a pin that's almost 30 years old).

You should be able to read cap values on the board to replace them. However, sometimes it's recommended to go higher than original for reliability. I know great plains sells cap kits, doesn't look like they sell it for data east (unless I suck at searching), example:
http://www.greatplainselectronics.com/proddetail.asp?prod=BALLY-S51E-KIT

Even if they don't sell the kit, you can search on mouser or digikey for these same parts