The tape mechanism may just sound louder in the case because it is using the whole casing as a sounding board. I've ordered a set of belts for mine.
Joysticks: The Spectrum (16K / 48K) didn't originally have or support a joystick so this was left up to a third party, Kempston, to come up with an add on interface and it was quite successful so a lot of the early games which support joysticks ONLY support Kempston. It was a a while later that Sinclair launched the 'Interface 2' which was a twin joystick and cartridge port - of course they didn't make it compatible with the Kempston 'standard' so pretty soon after that games were having to support both the Kempston and Sinclair standard for joysticks and offer that choice in the game's start menu. Fast forward to the 128 +2 and the computers are still using the 'Sinclair' standard for input from the ports so that is what you have to select in-game if you want to use the onboard ports, only on the 128 +2 there is also the additional problem that the joystick ports have been intentionally rewired to be non standard so you then need to use a converter cable to allow standard Atari-wired sticks to be used. If you don't specifically choose to use a joystick in the initial game menus the game will assume you are using keys by default even when there is a joystick connected.
Assuming your game lets you choose 'Sinclair' joysticks, a single player game will usually expect you to have the joystick in joystick port 1, not joystick port 2. Port one is counter-intuitively the right-hand port of the two ports on the side, anyone with a normally wired brain would expect that the leftmost one would be port 1 and the rightmost one would be port 2.
The good (?) thing about the Sinclair joystick standard is that it maps the joystick inputs to the keyboard numeric keys so if you start the machine up and go into (say) 128K mode at the BASIC entry point and move the joystick around, each action up, down, left, right, fire, should 'Type' a numeric character as though you are pressing a numeric key on the keyboard. If even that doesn't work then you may need to go over your joystick converter wiring again.
Not sure I can explain your sound problem at the moment, but bearing in mind that my TR4, TR5, TR7 were the wrong way around the whole time I was using the machine originally, I definitely remember the sound working, I had several games which specifically supported the three channel sound chip of the 128.
A quick sound test: The BASIC syntax for making simple beep sounds is:-
BEEP duration, note
For example,
BEEP 1,0