Is this a valid principle of curing HIV?
HIV hides in a number of known cells and unknown cells. Although most of the damage of HIV is caused by its infection of shorter lived cells, long lived cells which it infects includes anything with adequate receptors including memory T cells, dendritic cells, macrophages and some glial cells too. These cells are found widespread in the body from tissues to the blood stream. HIV also infects an as of yet unknown pool, which is thought to be the cause of rebound after HART (Highly Active Retroviral Therapy) is stopped. 

The constant parts are not exactly there. Every part of HIV can essentially mutate and does. In functional HIV particles (which surprisingly make up a tiny proportion of total HIV particles in an infected individual), the epitopes you mentioned are mostly constant. 

Destroying them is impossible as of yet and hence no cure for HIV. It is hard to destroy a particular protein. We could make HIV that doesn't make these proteins (it would then rapidly mutate until it did, or be non-functional to the point it was an awful vaccine not inducing a particularly adept CD8 response). Otherwise we could block either these proteins themselves or that which they bind. There is a group of individuals with the delta 32 mutation which blocks CCR5 preventing infection, however this is rare. For the most part most individuals are susceptible as CD4 and CCR5 are highly conserved amongst humans. We would benefit therefore from inhibitors of these proteins themselves, however due to glycosylation generally our antibodies are terrible at inhibiting the glycoproteins. Reverse transcriptase inhibitors exist and are in fact very important as part of HART.

HIV came from chimpanzees. In their natural host the HIV equivalent (SIV) doesn't cause immunosuppression. HIV doesn't infect quite a few primates due to restriction factors which do not exist in humans or do not target the same things. Then of course there's the matter of the lack of the correct receptors. There's ongoing research if there's a chemical that does cure HIV however there isn't a subset of individuals where this occurs to study. HIV resistance is normally due to elite control of the virus due to specific genetic features that make the immune system target the best least variable, most deleterious (if HIV mutates them to escape) epitopes. If we could get everyone to respond to these epitopes that would be great (see a billion vaccine trials for HIV) otherwise if we could flush out the cells where it hides latently integrated (our current therapy and the immune system can only target cells where HIV is constantly replicating).