Balance boards are great for helping improve both your reflexes AND the strength of the muscles used to support the ankles, ligaments aside. They are also a ton of fun to just goof off on.
We have 3 different ones, an indo board, a bongo board, and the weird one (Drift) that Quirky came out with (that went on sale for $20 last year - that's why I have one when I already had the other two, I just can't resist a good deal on something that looks like fun, but it's not being sold any more). Of the 3, I would say the indo board is easiest, but the bongo board is more dynamic (it rocks front/back as well as rolling side to side) and I prefer it. The Drift is the more difficult of the 3 for me, but my kids seem to think it's easier - I think it's because they are all much lighter than I am and they tend to keep their weight over the stationary part, rather than in the middle.
Back to skates though, it's important that you be properly fitted no matter what you end up with. Riedell does make good skates, so don't immediately eliminate them as an option just because that's all that is available locally. ANY skating boots beyond the basic recreational-only models are going to offer much more support than any rental skate does, so you really don't need to concern yourself with buying "more" boot to compensate for your own perception of having weak ankles, you need to focus more on buying something that is appropriate for your current level/ability/strength, and that fits well - after your first pair, then you can better evaluate what you need in your next pair of skates. If you do get skates with leather soles you will also have the option to get your blades adjusted if they aren't in the right position for you - and since everyone has different feet and centers of gravity, this is a very common thing to have to do. Footbed support is also important, and some issues like pronation or falling to an inside edge constantly can be corrected just by supporting the foot better, not just by moving the blades.
Query makes a good point about skating in a boot that is "too stiff" for you. Many of us adults do skate in boots that are rated "above" our level, but our own skating history has shown us that we actually need the extra support, we didn't simply start off there - usually what happens is that we've broken down a pair of skates that fit correctly much sooner than expected (poor fit can also accelerate break-down though, so that is something to be aware of), so we moved up in strength for the next pair. It's actually better to be slightly underbooted than to be significantly overbooted... because the risk of boot-related injury aside, if you can't bend your ankles easily, a lot of skills are much more difficult or even bordering on impossible to learn since you will be fighting the boot that doesn't want to bend. As a beginning adult though, you can safely consider boot options that are a level or two above what is suggested for a child, but I wouldn't suggest anything beyond that. Most kids wouldn't be able to bend a Jackson Freestyle as a beginner for example, but most beginning adults do just fine in them - as long as they fit well. A really good test to tell if a boot is too stiff for you when you try them on is to bend your knees while standing - if you can't push your knees past your toes as you look down, or without your heels lifting up and rocking your foot forward, then the boots are too stiff. Likewise, if it's much too easy that may be an indication that you need a little more support and could try the next model up.