No, Rega turntables are not only designed for Rega cartridges.
Also note that their tonearms has great reputation and are used by many makers other than Rega. Rega sells OEM tonearms to many makers. You can use other cartridges on Rega turntables and tonearms than Regas own, and there are many in this forum that does that.
VTA is not adjustable on entry level turntables and tonearms, including Rega. But in real life this doesn't mean so much because the error will be so little it doesn't really matter if you are not an audiophile with golden ears. Sound is still better than decent. On more expensive Rega tonearms you adjust VTA by adding shrims to the tonearm. Thorens used to include extensions between tonearm and cartridge instead of VTA, in their turntable packages. Those extensions came in different thickness.
Also note that vinyl not always comes with the same thickness, and older vinyl is often warped so you get different VTA's for every song you play... or you get different VTA's with different records. But it can still sound highly musical and nice. So VTA is not really that important.
VTF - Vertical Tracking Force - and antiskating, are the most important factors in tone arm adjustments, plus the position of the cartridge in the tonearm (to get the stylus positioned right). All the entry level models I have named here allows for adjustment of those, so you can use different cartridges with them.
You do get distortion with vinyl, if you want to be as free from it as possible then you want a turntable with a linear tracking tonearm. Those are extremely expensive. Linear tracking tonearms reads vinyl the same was as they are written so you don't get an error this way. Turntables has non-linear tracking tonearms because it is cheaper and easier to use, and because the error is often not heard by most listeners. With non-linear tracking tonearms you adjust them to give best performance in the middle of the record this will give the least error on the whole record compared to if you set the tracking angle at the beginning or at the end of the record.
Inner groove distortion can be a problem with all turntables (except linear tracking ones), but some cartridges are more sensitive than others. If the cartridge is properly mounted in the tonearm and the tonearm is properly adjusted, then inner groove distortion shouldn't really be a problem.
As for Pro-Ject and hum - hum is because of poor grounding. Grounding can be a problem with turntables yes, all turntables, and entry levels are lesser isolated than more expensive ones. How bad this humming is, is dependent on several factors like the surrounding equipment and how you connect the ground. On my setup, connecting the ground on the amplifier ground-connector provided less successful than connecting it to one of the screws on the chassi to my NAD tape deck. So one may have to experiment a bit to find the best grounding to avoid hum.
My turntable sounds good, not only decent, and I haven't tweaked it at all... by many vinyl lovers loves to tweak. It is not essential to get a good sound, but it has become a hobby in itself to find the best tweak...
Distorsion, VTA, VTF, hum, rumble, wow, flutter, clicks and pops and blah blah... if you are only looking at technical specifications then vinyl is inferior to CDs. Still vinyl sounds highly musical, can have more opened and broader soundstage, sounding more lifelike, more real, having timing and presence. And this even on an entry level turntable. Don't get too much into the specifications, use your ears instead and listen...Vinyl is really simple actually, from my point of view.