I would use the larger size potentiometers - 24mm. All the makes of potentiometer are good ones. CTS is the brand that most US guitar manufacturers use. Volume potentiometers should always be log/audio taper types. With tone potentiometers, you have a choice between using log and linear pots depending on whether toy want a fast of slower acting tone control. With single coils, I'd normally choose potentiometers with a log characteristic. For Fender-style knobs you really need a split shaft with splines to guarantee that the knobs will fit on well, so https://www.musikding.de/Bourns-Pot-...0k-log-knurled
If you never use the tone controls, then it doesn't really matter whether they are linear or log.
You can check the general amount of bow in the neck by holding down the bottom E string at the 1st and 17th frets. This needs to be done with all the strings on and tuned. You can then look at the height of the string above the 7th fret. For a standard set-up, there should be between 0.5mm and 1mm of gap between the fret and the string. Any more than 1mm and the neck needs to be straightened (truss rod nut turned clockwise/to the right). If there's no gap, then the truss rod nut needs to be loosened (truss rod nut turned anti-clockwise/to the left).
Unless the truss rod nut is very loose, never turn it more than 1/4 turn at a time. Check the result, then wait a few minutes then check again (as it can take a little while for the neck to fully respond to the change in truss-rod tension). Check the amount of neck bow again and if necessary, make another adjustment (again 1/4 turn maximum) After any change in the truss rod nut position, check the amount of bow. Once you're happy, then leave the neck overnight and check again the next day, as sometimes the neck continues to move a little bit.
This may not fully solve the string height issue, but you need to get the neck bow set up correctly before knowing what the next steps to take are.