> My guess is you can change modes throughout the blocks of a song (verse, chorus etc). So for example the bass would play the Ionian (B minor in this case) in the verse and then change to Dorian in the chorus (C# Dorian)? And can the modes change constantly with the chord progression?

It seems like you're confusing two concepts, which is no doubt understandable in light of the material you've been seeing.

First, there's a concept of "mode" that is analogous to the concept of tonality.  Just as a melody can be in a major key or a minor key, it can also be in a Dorian mode or a Mixolydian mode.  This concept of modes goes back thousands of years, before the concept of major and minor keys even existed (that concept only goes back three or four hundred years).  This idea of classifying melodies by mode also extends to harmonizing these melodies.  Traditional analysis typically identifies a single mode for an entire piece or perhaps each section of the piece.  For example, you might say that a song is in E minor but the bridge or chorus is in C major.  Similarly, a song might be in E Dorian with a bridge in E Mixolydian.

The other application of this concept is that it is a convenient way of identifying a set of notes that sounds good in your solo.  In other words, it's a helpful framework for improvisation.  In this context, you might switch to a different mode for just a single chord.  If your E Dorian song, for example, has an F chord in it, for example, you're probably not going to want to play E or F♯ during that chord.  The mode describing the set of pitches you want to use at any given point in a piece of music may not be the mode of the piece.

> All I see about these modes uses C major scale as the Ionian mode. Can other scales be Ionian? For example, can B# (edit: B# doesn't exist, I meant to type Bminor. I apologize) scale be Ionian?

Other scales can be Ionian, but only if they're major scales.  B minor has a key signature of two sharps.  This key signature gives you D major, D Ionian, E Dorian, F♯ Phrygian, etc., including B minor and B Aeolian.

> I think I understood that modes change throughout a song.

Only the second application of the concept changes throughout a song.  The first application changes more slowly, if at all.

> Do they change with the chord progression? For example, lets take the C major progression: C-G-Am-F. Will the modes then go Ionian-Mixolydian-Aeolian-Lydian? If not, when do they usually change?

Let me ask this: suppose you're playing scales in this song.  During the C major chord, you play C-D-E-F-G-A-B-C.  During the G major chord, you play G-A-B-C-D-E-F-G.  Does it help you to think of the second scale as a G Mixolydian scale instead of as a C major scale that starts and ends on the fifth degree?  In the end it doesn't really matter how you approach it except to the extent that it helps you understand what you're playing and why it works with the given chord changes.