I have some old Kyocera gear from the late 1980's, mainly a receiver / amp and a tape deck. I threw the CD player away years ago and wish I hadn't. I need to get the receiver recapped and see how it sounds. It sounded fine back in the day.
Yes. Some players do this. Look for a feature called "normalize volume".
Alternatively, you can use a utility that will look at all of the tracks and normalize their volume so that they are the same no matter what player.
An L-pad would allow you to attenuate the signal while not changing the impedance seen by the amplifier. Yes, they could certainly help you with the squawker level if introduced at that point in the circuit. Did you change the tap on the transformer yet? That would allow you to adjust it somewhat with the hardware you already have. Maybe someone that knows more than I do will chime in here....
Using your phone and 3.5mm headphone to RCA means that you're limited by the quality of the amplifier in the phone. I suspect that the noise floor of that phone amp isn't good either.
You probably cannot run your phone at max. That's a powered output and you'll easily overdrive the input of your amplifier.
You might consider Roon + Tidal.