It's been awhile since I checked in around here and I see it's been awhile since you posted this, so if you haven't given up on the idea of getting tech info from this site for what is definitely going to be a very custom build, I'll throw out the little bit of general knowledge I have that might be useful to you:
First on the valves: 20R and 22R valves are different lengths. Putting 22R valves in a 20R head causes some valvetrain geometry issues but it can be done. It would be better to buy valves that are the correct length though. Since this is a turbo app, valve size isn't going to be your biggest concern anyway. There are oversized valves available for the 20R that are the same length as stock, they're just not quite as big as the 22R valves.
Pistons - You will probably have to have custom pistons made. In stock form domed pistons are required in the 20/22R to get the compression ratio up to a reasonable level because the combustion chambers are rather large. The 22RE used a smaller combustion chamber design and therefore was able to use a flat-top piston (which is more efficient). There are lots of variables in a turbo build and one of them is your compression ratio. Most turbo motors that I know of don't run compression ratios above 9:1 and most are lower than that. The higher your compression ratio the less boost you can safely run and vice-versa. The tradeoff is that low compression ratios make for a less responsive and torquey engine just off-idle before the turbo begins building boost, but the ability to run higher boost pressures means more peak power. You'll need to acquire some specs and do some calculations to see what kind of compression ratio you'd get using the 22RE flat tops with 20R heads. Might be just fine or might be way too low (7:1?). Stock type domed pistons will push your compression ratio up into the mid 9:1 range or higher, which is going to be too high to run hardly any boost.
Turbo - you really need to do some research on how to size a turbo for a particular application. Proper turbo sizing is critical to performance and for a street driven vehicle it's generally better to go a little small on the turbo than too big. A small turbo will spool up faster and make more boost in the lower RPM ranges where you actually drive on the street. A big turbo will be "lazy" and not make boost until high RPMS which may be way beyond the limits of your shortblock or outside your normal driving range. You will really need to find the specs on that Volvo turbo then learn how those specs apply to a particular engine displacement to see if it makes any sense to use it for this build.
Other thoughts - Fuel injectors: You need to properly size the fuel injectors for the application. It is unlikely that ANY Toyota factory fuel injectors will be sufficient (I could be wrong here - it's possible the RTE injectors were oversized from the factory, but I wouldn't assume that under any circumstances). Undersized fuel injectors will run lean and destroy your engine.
It sounds to me like you need to do a lot of reading about turbo applications and how the various factors such as compression ratio, engine displacement, turbo sizing, and fuel injector flow rates all interract. This info is out there on the web but it will probably take awhile to get a clear understanding of how to do what you want to do. This site is great for information on suspensions, common engine builds, common engine swaps, and stock engine building, but custom stuff like you're planning doesn't get done much around here.
Honestly, for the $$$ spent and the amount of time you'll put into trying to figure this out with no guarantees of success you'd be a lot better off just swapping a late model V6 or even a Chevy V8. They are plenty reliable and will give you all the power you need. A custom turbo motor is going to be far less reliable than a well built NA large displacement engine.
