rests and/or slashes misbehaving?

a.) No. The slash notehead is part of the "musical font" used for notation, that remains true for tab as well

b.) Correct and (somewhat) intentional. If the slashes are created in voice 3, then they'll be above the staff. Test this out by entering a note in voice 1 & 2 and then issue the Fill with slashes command.
If you wish to fix the horizontal position of already existing slashes, you'll have to select them and adjust their "fix to line" property accordingly, using the inspector.

c.) I think the cause here is that the "slash" used the D3 note on the same string as where you already had notes in m5. To accommodate the slashes, the existing notes on that same string were moved downward two strings (not one, because a note already was there). The moved notes have an identical pitch for me, so are not "wrong" at all.
Though I would consider this behavior to be unlikely the expected user behavior and propose you enter a feature request to get this changed.

d.) This only works "fine" in m11 because you already have 3 quarter rests entered there (possibly from previous slashes that have been deleted?). This becomes very easy to spot when you turn on rest visibility in the Advanced Staff/Part properties. All following measure still have the default measure rest in them.
The expectancy here though is not to enter notes "by beat" (often enough you need different durations anyhow) but left-to-right, just as how you'd read them.
To enter a note on beat 2, you'd need a rest on beat 1; so first enter that (semicolon does that).
Have another look at https://musescore.org/en/handbook/3/tablature#keyboard with that knowledge in mind.

Part of your issues stem from trying to abuse the slashes feature for something it isn't quite meant to be used (as rests); which brings me to two items:

1. Why not enable rests visibility? Sounds like that'd fulfill your need by showing you exactly how long a duration the silence lasts..

2. How about using the following steps instead, to give you beat indicators over the entire score before starting note entry:

1. Select the entire (blank) score
2. Tools > Fill with Slashes
3. Select the entire (now slashes) score
4. Tools > Voices > Exchange voice 1 & 3
5. Change the "fix to line" value in the Inspector to '-1'

I think it would help to take a step back and ask yourself, why are you trying to add slashes? Simple slashes showing each beat are virtually never used in music that also shows actual notes - it's normally one or the other.

It sounds like maybe you are trying to create a special experimental style of notation where you use slashes to show where the beat is, to aid people in guessing the rhythms of the tab. Am I understanding correctly? If so, is there a reason to do this instead of simply showing the rhythm of the tab directly like publishers would normally do, using stems etc? I'm not aware of published music that does what it sounds like you are trying to do, and it's likely to confuse musicians reading it, because slashes normally mean something else.

Rhythms in tab are normally shown through other means, which you can get by switching to the "common" or "full" instead of "simple" tab styles, in Edit / Instruments. But if you wish to experiment with this new style, I'd recommend

As mentioned, using voices to place slashes above below works if you do that fiorst, before converting to slashes. But also, it's not meant for simple beat slashes like this, because as mentioned, that's not normally done inpublished music. The slashes above/below the staff are to show specific rhythms, not just counting beats. And that's why I think trying to do this is likely to just confuse musicians, so i don't recommend trying, For for the record, you can change the line used for slashes using the "Fix to line" sertting in the Inspector, as mentioned. This isn't recommended, and may not be supported in MuseScore 4.