This question was phrased such that it asks for opinions.
They should have rights.
This does not give humans any legal standing in cases of abuse. In the same way animals cannot have rights, humans DO have duties and responsibilities. Most US communities do not allow a person to have a poorly kept, extremely unkempt or dangerous residence; similarly, humans are not allowed to abuse animals to the point of starvation, thirst, disease, and/or death.
Cases of alleged abuse are normally handled by local law enforcement with the possible assistance of animal welfare agencies.
That depends on what you mean, are there any rights that are recognize and made into law? Not really, there are laws protecting very select animals, and they aren't very good ones, but sentient beings all have rights whether it is reflected in law or not.