1. It hurts to bend it a lot. For example, if I kneel and sit back on my heels. Should I do gentle stretching (that is, bend it) or not?
Reminds me of the Groucho Marx routine "Doctor, it hurts when I do this."
That squatting position is one of the worst for putting pressure on knee injuries-- you're maxing the flexion of everything you have left in the joint. You probably injured your knee recently (or just hit the low-level alarm on its worn-down cartilage) and your body is telling you that it needs some recovery/physical therapy. You might have compounded the damage of the previous injury or this is is a totally new problem. Hard to tell.
So yes, gentle stretching to warm up the muscles, but not so aggressively that you're stretching the ligaments & tendons. (You want tight ligaments & tendons so that the parts don't wobble around while you're walking/running.) Instead of static/passive stretches, try dynamic stretches of the activity you're doing-- walking before running, or using your wobble board. If you really want to get into the medical/physical details read
Thomas Kurz' "Stretching Scientifically".
2. Should I give it as much rest as possible or do some walking and even running?
Keep moving. Your quadriceps atrophy extremely quickly if you don't exercise the joint, and things will just get weaker & stiffer. You might benefit from 600-800 mg of ibuprofen before exercising (I use 800 mg because I'm 185 pounds). You won't feel the warning pain but you'll also minimize the swelling. There's no reason to run 10Ks or marathons, but low-impact activity should keep you in shape while healing.
3. Are my running days over?
Hard question to answer. Groucho would say "Then don't do that anymore!" Some people are built to run and have perfect form all their life. Others wear out. Do you need to keep pounding your feet on the ground and risk total knee replacement in your 70s? Or could you happily spend the rest of your life at moderate low-impact walking & biking?
An additional approach would be to add more knee exercises to your routine and see how your knees respond. I personally loathe these two exercises-- if you're not already doing them then they will challenge your quads & joint strength.
- Work up to 2-3 sets of 15x squats (with or without light weights) on a slow down four-count (femurs parallel to the ground) and a two-count pop up. Single-leg squats without weights are more challenging than double-leg squats with weight-- the idea is to practice control/stability along with building muscle.
- Hold your arms straight in front of you and slowly lunge to a full extension (front knee directly over ankle). Once extended & stable, twist your arms/torso as if you're moving a volleyball from your left side to your right side and back in front. Slowly come up and walk forward to lunge the other leg. Move the beach ball again. 12 steps out and 12 steps back. Three sets. For variety, twist the volleyball back over your shoulders instead of side to side. Going slow and maintaining full stability/control is more important than speed.
When you're feeling strong at those levels then do the second exercise by staying in the lunge as you move forward-- keeping your head at the same height all the way through. Again the challenge is tracking your knees straight ahead instead of wobbling or weaving along the line.
There's also jumping rope on a bouncy surface like a mat or the grass, especially hopping on one foot. But personally I think it gets pretty boring after 5-10 minutes.
At physical therapy I also learned about balancing on one foot on wobble boards or thick foam mats. For extra fun the PT staff would have us toss medicine balls around while standing in that position.
Or you could just do more surfing...