Is the universe some sort of inflating bubble?

No, the Universe being an inflating balloon is a misconception. In reality, it is more like an infinite graph, which stretches into itself. The idea that the universe had a single origin point has raised due to confusing terms like The Big Bang, which makes it seem as though the Universe started expanding from a single point, but in reality it started expanding everywhere, into itself, unlike the balloon, which starts expanding from a single point.

The answer to your last question is yes, because the Hubble constant isn't really "constant": It's the value (which varies with time) of the Hubble parameter at any given time. In the equation concerned, the "constant" does serve as the constant of proportionality between the "proper distance" to a galaxy (which can, unlike the comoving distance, change over time) and its speed of separation from the observer. That speed is described as the derivative of proper distance with respect to the cosmological time coordinate.

You'll run into arguments about the terminology on any physics site: I just got thru a low-key one, here on PSE, that lasted for weeks. Generally, in English, speed is a rate, and velocity is a speed of travel in one particular direction or another, but...expansion is different, especially when it's the expansion of great expanses of something (space) that's present everywhere. Expansion can, nevertheless, be safely described as occurring at some "rate", where it's occurring at all. That spatial expansion is occurring is virtually certain: It provides the simplest resolution of Olbers' Paradox, and has been verfied telescopically with extreme precision.

The answer to your title question is "maybe" or "sort of". A closed universe is usually depicted (at least in pop. sci.) as a sphere, and an "inflating" multiverse is often sketched as a finite or infinite collection of them. In one of the many papers that Nikodem J. Poplawski has, between 2010 and 2020, posted on the Arxiv site to describe his singularity-free, past- and future-eternal "cosmology with torsion", he describes the shape of each of its local universes as a "three-dimensional analog" of the surface of a basketball.