Is their a particular term for "standing" while cycling, and how is it done? I've never been able to manage it, nor for that matter, flinging my leg over the saddle while the bike is still in motion, when the ride comes to an end.
You can use the term "posting" to describe lifting yourself off the saddle when riding over bumps/potholes. It doesn't necessarily mean standing all the way up, but rather "floating" above the saddle so that your weight is on the pedals, not the saddle. This distributes your weight a little more evenly between both wheels, which can lessen the damage to your wheels on impact.
I wouldn't expect people to know this obscure term, but, regardless of the nomenclature, it is a very important skill to learn. Many people would benefit from riding more dynamically, that is, learning to shift their weight forwards and backwards, weighting and unweighting each wheel. The idea is that when your front wheel hits a pothole, you have shifted your weight as far back as possible. Then, by quickly throwing your weight forward, you can un-weight the rear wheel as it rolls over the hazard. Not only is this more comfortable for your butt and back, it puts minimal load on the wheels for the impact, which will help keep them true longer and reduces your chance of a pinch flat.
It can be called "standing sprint" if used by a road race bicyclist during a race/training. At MTB, it can be associated with "attack position", where one leaves the saddle momentarily (or for quite prolonged periods of time in fact, thanks @mattnz) to not be thrown out of the bike by the saddle launching the rider into space.
For regular commuters, Sheldon Brown considers the need to stand up while pedaling as a sign of a bad technique or poor bike fit:
If you find yourself standing to accelerate, on level ground, it is a sign that your gear is too high or that your saddle is too low.
Also from there:
Standing pedaling doesn't make you any faster, except in the very short run.
So, check your gearing technique and bike fit.
It looks like you aren't asking about standing and pedalling as such, but just standing at all? And your mention of not being able to mount and dismount by raising one leg over the seat just confirms this.
It is worth buying a bike that suits your mobility and balance. Bikes traditionally aimed at women in dresses have no top tube so you can "step through" so not being able to step over is not the end of the world.
Being able to stand while cycling is just a matter of balance and practice. If you are freewheeling, holding the seat or top tube between the legs gives a lot of balance and control, but if your hands are on the handlebars it should be straightforward to stand on the pedals (or even on one pedal) and balance just fine.
Just put in practice - when sitting and travelling gently downhill (so you don't need to pedal) try standing up. You can sit down as soon as it feels unstable. Once you have perfected this, try pedalling while standing.