Does it erase first, then write?

Apparently not. I guess since it's binary, it needn't worry about 'ghosting' of old data.

Then what's with the "erase" head[s]?

Those erase (maybe 'crop'?) the edges of what was just-written. "tunnel erase". Apparently the read/write head is slightly wider than the data-track.

Are read and write on the same head?

For floppies and older hard drives, it seems so. Though that same head may have separate windings for read and write, the actual "horseshoe" of the two electromagnets is shared. Thus, there's no physical displacement between read and write.

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Here's some references:

http://www.pcguide.com/ref/fdd/constHeads-c.html

http://www.retrotechnology.com/herbs_stuff/drive.html

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Random unanswered Qs:

How does it determine which polarization to write, when?

Best bet: it just toggles... starting randomly, or maybe always starting the same. But the actual polarization at any position is irrelevant, as zeros are represented by no change in polarization, and ones are represented by a change (flux). Since individual bits are never modified, and everything happens in sectors, and since sectors always start with a long synchronization sequence, it really doesn't matter what polarization it starts writing with. 

It's *highly* doubtful, to me, that it starts with whatever polarity was last-read... that'd be somethin'. But I'm pretty sure it takes a bit to switch between read and write, and by that time the head's already moved to the next bit. Further, it'd be plausible the old data would be more weakly-magnetized than the newly-written, even if the same polarity, which could be mistaken by the read-head/circuitry as a change-in-flux (one).

(This doesn't help my sectorless-single-nibble-rewrite ideas, but that was a far-aside)

BUT WAIT!!!

The read-write head is, essentially, a horseshoe-magnet. The media is written in the gap between the two sides of the horseshoe. Horseshoe magnets have *BOTH* North AND South poles.

A slightly different observation/question was answered (see Head Gap Width at: https://en.m.wikipedia.org/wiki/Tape_head) that the moving media receives the polarization from the trailing edge of the gap. 

(Ahhh! That might help explain why it needn't be erased first, it receives the opposite polarization at the leading-edge and gets repolarized at the trailing edge.)

But, then, when the write process completes, won't there be a flux-change inherent?

Again with the fact of sectors written in one single pass, and a differently-defined 'gap' of repeating data written at both the beginning and end.

BUT, this may have some relevance to PWM-nibbles.... to be pondered.

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What about magnetic monopoles? If the media is polarized by the trailing-edge of the horseshoe-gap, which is, say, South. And, say, the disk spins 1/4in, causing a North(?) polarization on the media for 1/4in... where is the corresponding South pole on the media? Wouldn't it have to be large enough, or strong enough, to match the total flux of that 1/4in North pole?

Opposite side of the magnetic coating? (rather than the plastic)? But, nah... because Perpendicular Recording is a much newer feature... (TODO look up). And am pretty sure the head-gap (the two sides of the horseshoe) is parallel to the track... (gah! is Perpendicular Recording perpendicular to the track ON the track's surface as opposed to perpendicular to the surface? TODO: LOOK UP!)

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https://en.m.wikipedia.org/wiki/Perpendicular_recording

well shee-it... this can't be right... for longitudinal... then a flux-change would be measured at every head-gap-width *except* where two 'magnets' are butted against each other at the same polarity. If that were the case, we'd need separate read/write heads, with different gap-sizes, for different recording densities... and what about spinning?

Holy schmoly... dig this:

The larger the magnetic region is and the higher the magnetic coercivity of the material, the more stable the medium is. Thus, there is a minimum size for a magnetic region at a given temperature and coercivity. If it is any smaller it is likely to be spontaneously de-magnetized by local thermal fluctuations.

This might not bode so well for increased-width PWM-nibbles, which was an idea for experimental purposes...

Nonono, "Minimum size," not maximum... sheesh, brain.

But, maximum seems relevant, too... per my earlier thoughts.