There's some variation on 390s so knowing what you really have is the first riddle you need to solve. Is it a factory 4v?
If the engine is on a stand its certainly not unreasonable to pull the pan and take a peak. As mentioned before gaskets are cheap. What you don't want to find is a set of bearings that look like a hundred year old man running a marathon. Also a new
oil pump shouldn't break the bank but as mentioned, once you start down that road its a bit of a rabbit hole. Even a stock
oil pump is pretty easy to clean up though. Look at the bottom plate, you can hand lap it on a flat surface, the wife's marble kitchen counter tops are pretty good for this. You'd also feel pretty stupid to find out after that fact that the pick up was partially blocked or just really dirty. That alone is all but guaranteed by the looks of what you have. It is completely possible to freshen up the bottom end on a budget though. At a minimum, avoid a significant loss. I can mail you an old cam I have, it would come in about 5 different boxes and might have bits of connecting rod impregnated...
As far as heads go, lots of good advice here. Clearly blykins has lots of experience. While bigger valves don't always make more power, it's a good bet in this case the TFS heads would be a real improvement. Pay attention to the
oil drain back.
Aluminum heads tend to run a little cooler, and can be more forgiving on detonation. They're a lot lighter, always good. They are shiny and have a better smile to mile ratio and will serve you a well as long as you don't grenade the engine too badly. Again, rabbit hole warning, you're not going to feel really great installing aluminum heads and not getting a new intake manifold, and a cam, and new rockers, and a new distributor, and a new double roller, and you might as well go roller lifters with the new cam, and a new carb, and wing nuts...
One thing to consider is installing TFS heads on a stock engine, which is fine, is that those old rings are gonna get a little more work than before. More heat, more pressure. They may (will) start to show their age. Throwing more power at a stock bottom end in my experience has much of the time lead to more issues.
But if you don't mind running a smoker, keep an eye on the plugs, replace them a bit more often, you may need to run a hotter plug which can lead to other problems, you can hedge a little by backing off the timing 2º or so. Its a tough thing to throw a power adder like a TFS head on then detune, one step forward, two steps back etc..(the one step forward being much bigger in this case)
The TFS heads will probably out last the bottom end. Those heads you posted pictures of at a minimum should get cleaned up. Looks like enough crap in there to potentially get into the rings. You just don't know. Again, it's already on the stand. You could hang a valve and really be disappointed later. A 3 angle valve job is a very decent performance gain and any basic shop can do this. Take a clean up pass on the head surface and call it macaroni. Yes you're going to lose a week or so depending on how much time you put into it.
390's are can be good engines. Bone stock it could be a good driver. Depending on how well worn the engine is....when in doubt a 250 shot of nitrous oxide is a good way to find the weak link. It comes under the heading "watch this"....it won't fix things but it will end the suspense.