Only orthopedic or rheumatologist if hypermobile joints continue to loosen, different parts of affected joint's cartilege wear down? Why or why not?

See details. True hypermobility syndromes often cause issues with chronic joint pain either from the loose-jointedness or an accelerated osteoarthritis.
Usually not. In some cases wear will happen...Notably subluxing shoulders and kneecaps. However in most cases only risks are of other issues like worsening instability if repetitive minor injury such as recurrent shoulder or patella dislocations . Most "loose joints" cause no symptoms or symptoms lessen with time.
We stiffen with age. Thankfully, most hypermobile joints stiffen naturaly with age. It is unlikely that an ything other than physical therapy for muscle conditioning is warranted. Over time irregular wear and tear (joint arthritis) can occur, and is best evaluated by your orthopaedic surgeon.