Absolutely! With today's technology almost everyone is a candidate for dental implants. You will need to have a consultation with an oral surgeon and a dentist to evaluate the quality and quantity of your bone. They will then be able to present options for restoring your smile and chewing function.
Usually it can be fixed. Dental implants help preserve facial bone and lip support. A consultation with a board certified oral surgeon will help answer your questions.
In my opinion, a sliding genioplasty is the best option. First of all, shaving down the chin is not a recommended procedure. The option of getting lips enlarged would only camouflage the actual skeletal discrepancy of the chin. I believe in fixing the problem instead of hiding it with something else. A sliding genioplasty, while not to be taken lightly, is a relative minor procedure with few risks. I would highly recommend looking into this option.
Jaw surgery could absolutely fix your open bite. Generally, misaligned teeth are caused by one or both of two things: teeth in the wrong place or bones in the wrong place. Obviously, the easiest way to correct this is to wear braces. If all or part of the problem is your bones being in the wrong place, then braces can only mask or camouflage the skeletal discrepancy. Jaw surgery can correct the skeletal position of your facial bones and speed up the correction of your open bite.
It appears you have a dental infection with abscess formation. This could be due to dental decay, periodontal disease, or both. Unfortunately, you need to seek dental care in spite of the fact you do not have dental coverage. This problem will not resolve itself and left untreated could cause more damage which will require more treatment. Treatment with antibiotics will only treat the symptoms, not the cause. Dental infections in the upper jaw are also noted to spread to surrounding areas including eyes and brain.