My left knee is screwed up. I can?t even squat close to failure in my thighs at high or low weights, even using a brace, without knee pain kicking in. Leg extensions do the same. I have a long history of trouble with that knee, and knee problems run in the family, so I need to jump through the insurance and medical hoops to straighten this out.
In the meanwhile should I keep doing lunges , leg curls, and straight-leg calf raises to work the rest of my legs, or should I just leave it all be until a doctor has looked my knee over? Are there any knee-friendly thigh exercises to replace the squats/squat press?
Well, I think you may have a hard time getting feedback with the information you have provided. Pain can be widely varied. There are various types of pain and also various tolerances to pain. Certain pains are just indicative of a good workout while other pain spells trouble. Giving some detail as to the type of pain (burning, piercing, throbbing, stabbing, dull, etc.), the intensity of pain (mild to severe) and your exact tolerance to pain (low to high) will help.
In general I would avoid exercise that causes pain until you get the knee checked out. If you can do lunges without pain there are a wide variety of lunges available that can help build up your legs. If you can do lunges, can you also do stepups? If so, that opens up another set of leg exercises. If you can do deadlifts that opens up a whole new set of exercises.
You can easily find all kinds of variation of these exercises on this website. There are also a lot of different variations of squats. Do all the variations hurt, or just certain ones. Don’t injury yourself. Work what you can and listen to your body. Good luck.
The pain is a sharp pain in the inner/upper area of the kneed just under the kneecap. It begins after I bend my knee beyond a certain point and eases once my leg has straightened out again; with each repetition I can bend the knee a little less before the pain kicks in.
I know that there?s more to this than just my technique because I have similar pain that starts after about fifteen minutes of running and it also sometimes hits me (mildly) when walking down stairs. Until recently the pain was always temporary and mild, but when I started pushing my thigh exercises things got nasty.
Of course when I tried to talk to my insurance provider about a doctor today it turned out that they canceled my policy without notice back in March, which is odd given that I have not seen a doctor in four years. Hopefully that nonsense can be resolved promptly 
I?m just getting into lunges and deadlifts now, but I?m having to ease my way into the lunges because my hamstrings are adjusting to the stretching. I?ll look into step-ups.
From your description it sounds like it could be something pretty serious, but I am not a doctor so it is hard to say. I feel your desire to workout. That is admirable, but make sure you don’t hurt yourself. You may want to put a little more focus on building up your upper body and keep the volume on the lower body down.
You will probably have a battle on your hands with the medical insurance company. The only thing they are really good at is collecting fees. You may need to get some legal advice. I really have never had to work around a bum knee like this, so I am hoping others on this site can chime in and give you some good advice. Stay in the fight and good luck.