Rheumatoid arthritis, so far as I'm aware, does not go hand in hand with hypermobility. The link between the two is FAR more tenuous than that, as one relates to over-stretchy connective tissue (hypermobility) and the other is an immune system dysfunction (RA). Hypermobility alone is very common in the general population, RA is not nearly so common and so far as I'm aware, you can have RA without being remotely hypermobile (although I really don't know much about RA) and can most definitely be hypermobile without having RA.
What IS frequently linked with hypermobility, though, is pain. Pain is frequently found with hypermobility for various reasons - one being that if you bend your joints too far the wrong way, you are more likely to damage something, dislocate something or sublux something; another is that, for some reason, hypermobile people, whilst they can bend their joints further than others before it starts to hurt, find they take longer to heal if they do manage to damage their joints. Hypermobility is often linked with earlier than average onset of osteoarthritis (ie arthritis cause by wear and tear rather than an overactive immune system attacking itself). It is actually also associated with an increased tendency towards osteoporosis, easier bruising, and even finding that local anaesthetics do not work... but then plenty of people have hypermobility and only ever find it an advantage to them, eg in sport.
As others have said, you can't cure hypermobility, the aim is to ensure that your hypermobility doesn't expose you to the risk of damaging your joints. Strong muscles supporting the joints and compensating for over stretchy ligaments and tendons help, hence physiotherapy being very helpful (physio also helps train you to use your body in a safe way, rather than inadvertently moving around in a way that harms your body). Special supports can also help for joints that are too unstable, and learning to pace yourself so that you don't put too much repetitive strain on the same joints all day every day.