That case that you described is not the norm today. People with HIV and AID live very long and normal as possible lives. Yes they still get the same illness and injuries as anyone that didn't have the disease. HAART meds have really improved in the last 10 years.
I took care of someone with end stages of AIDS and they were very frail and thin. He was fully dependent with care in the nursing home, and he couldn't walk anymore. I had to use a machine to lift him in his wheel chair. He would shake, and his eyes would move back and forth at the same time, and we had to feed him, and he did get very skinny towards his final days. I want to cry evey time i think about it because he was so young and the worst death from disease i've ever seen, and i've seen alot! So Yes you do loose weight, not sure on exact times, but for sure towards the end of life.
NO, you don't continue to lose weight. You lose weight just like anyone else that would lose weight with an illness. It depends on the person and their eating habits. Yes, you can get fat and have AIDS.