They aren't using paste any more. Since 11th gen Intel's desktop CPUs are soldered, if I remember correctly they use some kind of indium alloy, and so does AMD.

Edit: All desktop CPUs since 11th gen and some 9th and 10th gen according to Intel

Hopefully it'll be in auch better cooling situation than your laptop.

You need to continuously supply it with more coolant. I don't know what the thermal capacity of a tank of liquid nitrogen is, but whatever it is, it will eventually run out and you will either need to condense more of it or acquire more from someone else.

You'll need to keep it pressurized or else you'll lose some from it just being in a room that's a safe temperature for you to be in. Even then, you'll still lose some to that when it's allowed to vaporize to dissipate the PC's heat. Unless you use a closed loop system, but you'd still need to dissipate that heat somehow while keeping it under dangerous pressure levels.

Phase change cooling systems can still get quite cold without some extreme liquids. Think like ACs and freezers. You can still get to the point where you need to consider potential condensation with one of those, though a room AC system will still cool your PC because normal cooling depend on ambient temperature.

As I understand it no, not without a large sum of money for a custom solution. They would probably need 2 phase cooling to reliquify the LN2 or helium.

Not quite as good, but some brands use to sell cases that could go below ambient using basically an air conditioner: https://www.trustedreviews.com/reviews/asetek-vapochill-xe-ii-refrigerated-pc-case