What verb or phrase do you use to describe the situation where someone is sitting in a rolling chair and they push against the floor to move around.
Other options are scoot (as in "(kick) scooter") or skate (as in skateboard).
Of the two I'd prefer to use the former, for example, "In one old job I could scoot my chair to my boss's desk and turn it to face the right way in one push" (which is true, though not acceptable in all workplaces).
I would use the word slide. If being in a chair isn't obvious from context, you'll probably need to make it clear.
- Let me get a closer look. I'll slide over to you.
- The boy got hurt after sliding quickly around in a rolling desk chair.
You could also use push as you did in your question:
- I'll just push myself over to the conference room.
These all are informal, but mostly because it's a fairly informal topic of conversation to begin with.
There are a number of possibles, but there probably isn't one specific word that fits particularly well. Here are three possibles, but there are probably a lot more.
In a small boat, if you are using an oar on each side like the picture, it is called "sculling." So that might work for using both feet.
If the person was using one foot only, it might be "poling."
If they are using furniture by grabbing it in addition to pushing with their feet, you might call that "crawling."