It sounds like a daunting task, but restoring your engine to factory (or slightly better) specifications isn't terribly difficult. We're going to break down this section into two main categories, Mechanical Performance and Visual.
We're basing this section on the general maintenance of factory performance specs. In other words, you're not trying to soup up the car or squeeze every last possible horsepower out of your engine (for that, see the Modifying section). You'd just be rebuilding or replacing for stock specs, and no more.
The good news is that nearly everything is still available in one form or another for your car's engine, no matter which version or fuel delivery type that you have. Ford produced so many 3.8L V6 and 5.0L V8 engines that rebuilds and bare blocks are readily available. The same goes for cylinder heads, camshafts, rockers, distributors, etc. Basically, if your engine is tired and needs rebuilt, there should really be no issue finding correctly specced parts for it. (The 2.3L turbo-4 is a little different; we'll discuss that later).
And therein lies the big question: to rebuild, or to buy an already remanufactured engine? That is usually a question that's answered by your budget, but that aside, it's all about convenience and time. If you have the time to have the engine removed and rebuilt by a quality machine shop, then by all means, that's the way to go. However, you might be under a deadline or you might have a tighter budget, and that's alright. Pre-rebuilt short- or long-blocks can be cost effective, and you usually will get a decent warranty with them as well.
For either situation, you will likely see a very slight bump in overall displacement, which is common. A standard rebuild will see cylinder bores honed out at .010" or .020" over, just enough to clean up the cylinder walls and get them evened out. Stock pistons, if still good, can be reused and fitted with slightly larger rings to make up for the larger gap. Or, if needed or desired, new pistons are available at .010" or .020" over. It's not a huge difference in displacement and is safe for cylinder wall thickness. Also, bear in mind that a .010" over block can usually be bored out again later on, if need be. If you have any questions or concerns, you can always ask a competent engine builder for advice. Just FYI—we aren't big fans of having the cylinder block sleeved, as that can cause a multitude of issues. Please try to avoid that at all costs.
Stock-spec camshafts are still available. This will ensure that your engine's computer will perform as expected. Anything over a mild stock cam upgrade can result in timing and fuel delivery issues, so be very careful here. And as always with a rebuild or remanufactured engine, be sure to insist upon a brand new double-roller timing chain to ensure longevity and consistent performance.
Similarly, cylinder heads are available in plentitude and we would highly recommend considering new heads instead of used ones. You can get remanufactured heads, and that's fine also. But new casting heads, particularly with the aluminum 3.8L V6 heads, are much better because they're straighter and can be engineered to resolve some factory flow and cooling issues. They will appear as stock, and perform as stock, but be brand new with a warranty…and there's no greater peace of mind than that. Some are sold bare while others are fully assembled. It's up to you as to which will work for your situation, but fully assembled heads drop right on with no transferring parts from the old ones, save for the rockers and pushrods. That might be a wiser choice for you in the long run, even if they're a bit more costly. No affiliation but we've had very good luck with heads from Clearwater Cylinder Head, and they're very affordable. Please see their site or call them for more details.
Your intake(s), throttle body, injectors, etc. can all be reused if they're still good. Same goes for the exhaust manifolds. All of your accessory brackets and pumps will bolt on (but make sure all the required threaded holes are in the new block and/or heads, or you'll be drilling and tapping).
Now about that 2.3L turbo-4 engine: you're either going to have an easy or difficult time finding parts, depending on what they are. A lot of non-turbo 2.3L parts just won't work, even though that engine was way more plentiful. However, there are plenty of other options out there, including performance shops like Esslinger, websites like NATO, and many Facebook groups that are all dedicated to turbo 2.3L performance. If you need parts, those are the places to look. Keep in mind that there may not be a lot of new part options but used parts seem to be available more than you may think.
If you're planning to compete in regional or national shows where as much of the car needs to be stock-appearing as possible, they you're going to need to dedicate a lot more money, time, and patience in keeping the engine and surrounding engine bay looking factory fresh.
As time goes onward, finding technically "correct-looking" parts for your engine can get a little more difficult. This means, correct dated hoses, factory-style wiring looms and tape, correct markings, etc. Let's face it: these cars, as luxury-oriented as they tended to be, were also common as daily drivers, so parts were replaced often. The good news is that a lot of other Ford vehicles shared similar components with our cars, so the parts themselves are generally easy to find: water pump, throttle body, EGR valve, etc. However, to be technically correct, you'd need to shell out more money for genuine Ford parts if they're available. And that's the trick.
Ford generally does not manufacture or stock parts for very long, even with new cars, so dealerships probably aren't going to carry most new parts for our cars any longer. Some exceptions are heater cores and radiators. So you'd be looking for NOS (new old stock) parts which therefore means finding the correct part number and hunting from there. It's not easy to do this, folks. We know of someone that's been looking for NOS dated belts for over a decade. Dated hoses are also tricky. Some of them have dates but they are a decade or more after the car's model year, which means there's no hiding the fact they've been replaced. However, as long as you have a dated hose that came from Ford, generally that's acceptable as a replacement part and you shouldn't get penalized. That same philosophy goes for other high-demand, often-replaced parts like belts, other hoses, and especially the battery.
Yet some parts can be had with either a Ford logo or part number. Usually they are smaller mechanical parts, such as the idle air controller (IAC), idle bypass valve, EGR valve, MAP sensor, and so on. Your local parts store might just be the ticket for these, as many of them used older Ford cores for remanufacture. This is going to require a lot of pulling parts out of boxes to visually inspect them. But you might get a good payoff in the process.
Also, don't forget that some Fox Mustang restoration companies such as LMR, Fox Mustang Restoration, and CJ Pony Parts carry some new parts with more or less correct date stampings and should be acceptable for originality. Especially for the 5.0L V8 engine, this can be a blessing. Be sure to compare your stock part with the offerings from these companies to make sure they're going to work for you.
So is it ever okay to put on a used part to replace yours? That really depends. We liken this analogy to installing used tires: sure, they'll work for awhile but eventually they're going to fail, and usually at a very bad time. But you may also have no other choice if the part you need isn't available new. So long as it doesn't impede performance or increase the likelihood of failure, you might just be alright.