Does high uric acid level damage joints?

Indirectly: High uric acid levels can cause crystals to form in the joints, causing the pain, joint damage and other problems in gout.

Not by itself: High uric acid, by itself, cannot damage joints. If and when it causes gouty arthritis can joint damage result. This type of arthritis may manifest in many different ways, from the solitary attack to multiple attacks and chronic arthritis over time. High uric acid, again, may be harmless, or it can also cause kidney problems, so get a urinalysis and kidney function tests to be on the safe side.

Possibly: It is possible but not necessary. If you only have hyperuricemia without gout or tophi formation, then no. If you develop chronic tophaceous gout, the tophi (large collections of crystals made from uric acid) can cause joint damage and erosions. In general, it is not recommended to treat elevated uric acid levels unless there is another medical problem like gout, heart failure or renal disease.

If gout occurs: Uric acid does not damage joints. But, if it gets too concentrated (>6 in blood test) it forms crystals that the body can "notice" and this triggers redness, swelling and pain mainly in/around joints. This is gout, and it starts in about 5% of people with a uric acid of 9 (per year). Recurrent gout flares cause joint damage which can cause persistent pain.