just so make sure I'm understanding this I could use some .303 dies to resize brass fired from other guns, then use these casings and just resize the neck after firing them from my gun?
WRT your first -
NO!! which means "It depends' which means 'Sometimes-But-You-Can't-Depend-On-It'. It hinges mainly on what dies you are using.
When a cartridge fires, the pressure inside the chamber forces the case to expand in every direction. It's minor (1/100ths or even 1/1,000ths of an inch) but it's there. It's definitely a Good Thing because it provides rearward obturation - seals the chamber and prevents incandescent high-pressure gasses from combustion from rearranging your iris.
A fraction of a second later, the pressure drops and the case contracts, just enough to allow itself to be extracted from the chamber. But it has not - not - returned to its original size.
A die that only resizes the neck (like the Lee Loader) leaves the main body of the casing the size it was ejected at. If you just neck resize and try putting that into your own rifle's chamber, it will fit.
If however, you are 1) using brass fired in another rifle and 2) if that rifle has a chamber just a hair larger than that of your rifle, then it will not - not - fit. And tolerances in firearms manufacturing are such that this can happen very easily, particularly if the other rifle is a bit older.
Now, if you use a die that resizes full-length, the casing will be squeezed down small enough to fit into any chamber. To do that however, you need a reloading press, proper dies, etc.
So - using a Lee Loader, no, you cannot depend on being able to use cases fired from other rifles.. Using a more expensive setup, yes.
And as you said that there are too many factors to determine how many times you can reload a case is there any approximate amount if you had to guess some number range, 2-5, 5-10?
That too depends. It depends on the quality of your brass. It depends on the size of your rifle chamber (if it's larger and you are full-length resizing, the brass is being worked just that much more). It depends on the loads you are using. It depends on a bunch of things. A .303 with normal loads, easily five times, probably more.
Just trying to determine what the saving may be like. Local CT sells new for ~$30 for 20, so >$1.50 / round.
Generally, the most expensive bits of a cartridge are the bullet and the casing. Reusing your casings is the biggest money-saver possible in this context.
I haven't bought .303 bullets lately, but let's say that they're $40 per 100. That's 40¢ per reload. Primers are, call it, 5¢ apiece. Your powder is, say, $40 per pound (7,000 gr) and you use (picking a number) 50 gr per round, that's 140 rounds per pound or 29¢ per shot.
(Note that I've not done precise research so please don't hold me to the numbers - they're in the right time zone, I think.) You've already paid for the brass when you bought commercial ammo, so you can call that essentially 'free'. Total cost for a reloaded round is therefore 74¢ - half what you are now paying.
Now you have to factor in the cost of the reloading setup you buy. Reload 20 rounds, you lose. Reload 200 rounds, you save. Reload 2,000 rounds, you save a bundle.
Plus which, you can tailor the loads to suit your needs and your individual rifle. (Easier with a more complex setup.)