Re: Footwear on the mat
You need to ask your Sensei first, because they can be a bit picky on this matter at times.
Of course, wearing sneakers would be unacceptable (and dangerous too), but there may be valid reason for some other type of legit footwear.
For instance, you can get fungi - although they may say it never happened to anyone there, getting fungis is not a function of an unhealty mat but of subjective predisposition, so objecting that it never happened to anyone does not mean yet that it may not happen to you.
If you had that problem in the past, you can explain you prefer some kind of footwear or even produce a medical paper where it is stated that, since you had fungis in the past, it is advisable that you wear some kind of foot protection.
You could also wear socks, simply, if your Sensei agrees. Although it is argued that with socks you may slip on the mat (which may constitute a relatively valid objection), I wore tennis socks often and I never slipped even once.
An alternative may be swimming pool shoes, they are very fit around your feet and made entirely of flexible material, and slipping wearing those is out of discussion for their purpose is exactly that of avoiding slipping even on wet surfaces (at least as possibile).
We could elaborate further on the topic: for instance, you may need knee pads. I always wear them in the dojo, under my trousers so nobody notices and no one cares. They help enormously for all the times they crack tough on me with nikkyos or when i fall under an ikkyo and yet they fail to make me lay flat immediately (I prefer acting naturally: if you cannot throw or ground me, you won't ground me).
I also wear a tshirt (cannot remember right now the term oh my, ok under the dojo jacket so to say lol) which at times has been found objectable (apparently in some dojos if you don't dress exactly like anyone else they find it not ok, but I have no idea why a tshirt is deemed a problem by some)