Each state generally sets its own standards for HOAs. There are legal frameworks, insurance laws, etc. which are fairly similar across the country.
>If they have equivalent legal status to a government or a police department, are there constitutional rights which they cannot violate?
They do not have equivalent legal status to a government or police department. HOAs are limited in their powers by federal, state, and municipal guidelines. Every state has its own laws in this regard. There are also federal laws prohibiting certain bylaws, e.g. the Fair Housing Act prevent HOAs from enforcing racially discriminatory covenants.
>Can two competing HOAs exist in the same geographic area, and home owners are free to setup additional ones if the incumbents don’t meet their needs?
I don't know exactly what you're referring to, but generally speaking, nobody is forced to buy in an HOA neighborhood unless they want to, there are generally a multitude of HOA-managed (and non-HOA-managed) neighborhoods to choose from within a single town/school district/etc. The only exception might be things like specific lakes/small island communities, where the entire geographic feature is managed under a single association (e.g. for lakes they generally pay for the maintenance of the lake itself, stocking fish, removing trash, etc.)