Thanks for all your responses. I had a look at the FAQ. I was actually lurking around the forum for a while before, but it wouldn't let me into the FAQ until I made an account.
Knowing that my Dad's Luger had been in a fire, I checked the barrel very carefully to make certain it was still straight and that there was no warping. I wouldn't have put live ammunition through it if I had any doubt at all about the barrel. Going to the range, my only concern was if the springs would be strong enough to do their thing or how it would cycle. It sounds pretty risky to trust a gun that had been in a fire like this, but it's not like I took it to the range from the ash heap.
I'm pretty sure that when the fire happened the gun was contained inside a metal box, so that must have helped protect it from the full intensity of the heat. So the story goes... my parents were newly weds and they were living in a trailer while they were building their first house. My mother was pregnant for me (their first child) and their trailer caught fire before the new house was finished. They pretty much lost everything in the fire, (including an antique violin that had been in the family for a long time.) I remember my mother telling me that the trailer was cheaply made with cardboard walls so it burnt real good. The fire happened at night and they were in bed when it started. Our parents told us they had to rush out literally naked, and Dad said he could hear the ammo going off in the fire. It's an exciting piece of family history associated with this gun.
From my research into the springs, it seems that a lot of the notoriety for jamming issues was caused by a difference in the hotness of 9mm loads. American versus the hotter European and NATO ammo. If I understand correctly, when fitted with fresh springs to the OEM strength, Lugers will function best with the hot loads, and supposedly American standard 9mm wasn't enough to make the gun cycle properly. So on the vet bring backs, Americans outfitted their shooting Lugers to slightly weaker springs so that they would cycle with American 9mm. (?)
I'm starting to wonder if my Dad's Luger was civilian modified post war, on our side of the pond. I wish I could still talk to my Dad, I never even got a chance to ask him where and how he ever got the gun in the first place. I always just took it for granted when he was alive. Him and my Grand Father were really into these things, and now it seems I'm the last gun guy still alive in our family.
Dad's Luger seemed to cycle just fine with cheap domestic ammo, so I'm wondering if it was already retrofitted with lighter springs, or if they were just appropriately weakened in the fire. ( LoL

) -If I went ahead and put in new springs, would it become more picky for hotter ammo? Also, when I bought the bullets, the guy at the gun store offered me bargain steel case ammo. I declined, as I figured brass would be softer on the internals, but I was wondering... do any of you guys ever use steel case 9mm in your Lugers? ( -That's probably a stupid question, but I'm a shameless newb! XD )
Also speaking of springs, I still want to fix this spring for the action hold open. This seems like it should be a pretty simple and easy leaf spring to make, but in my research I have read some bad reviews of replacement springs snapping as they're installed.

Can anybody recommend a reputable source for selling replacements?
So I just put in the order for these reproduction grips:
https://www.amazon.com/gp/product/B01NAQHK32 It also comes with the little tool, that should make reloading the mag a little easier.
And again about those two flat head screws someone put into the back toggle... Certainly that's not OEM. Do you guys think that looks like something that would have been done on our side of the pond post war? That's what I am starting to suspect.

As for Rick W's comment... do I want something original looking or more of a custom shooter? Well I would prefer the original look, but as far as sights go I'll prioritize function over form. If there was an easy back sight that I could use where it was cobbled I would be fine with that. When I took it to the range I didn't test accuracy, I didn't even post a target. I'll do that next time.
Thanks again for all your responses.

Here are some more pictures... Looking at the hold open dealy when I dissasembled it immediately following my little range test. A pic of the ammo I used and a pick of my magazines. Cheers.