is '4 minor' sub-dominant?

Generally speaking, iv has a sub-dominant function, both in minor, where it occurs naturally, and in major, where it occurs through modal mixture. So, for example:

X:0
K:Cmin
L:4
Q:"C minor ('native')"
[CFA]2 [=B,FG]2 [CEG]4 || [K:C] [Q:"C major (borrowed)"] \
[CF_A]2 [B,FG]2 [CEG]4 |]
s: iv V♮7 i iv V7 I

iv can be substituted for bVII7, but it functions differently. bVII7 bears more direct (aural) relationship to viio7 -- sort of a "gentler" version, but still serving a dominant function. iv, on the other hand, lacking both 7 and 2, functions still as a iv. That is to say, bVII7 - I sounds (more) like an authentic cadence, whereas iv - I (or i) is clearly plagal.

X:0
K:Cmin
[B,DFA]4 [CEG]4 || [=B,DFA]4 [CEG]4 |]
s: bVII i viio7 i

Part of the reason the substitution works, however, can be seen more clearly when considering bVII9, which contains iv.

X:0
K:Cmin
[B,DFAC']4 [FAC']4
s: bVII9 iv

iv7 is perhaps a bit of a misnomer, because it's not really a seventh chord when resolving to the tonic. It's actually functioning as an augmented sixth chord.

X:0
K:C
[F_AC'_E']4 [EGC'E']4 || [F_AC'^D']4 [EGC'E']4 |]
s: iv7 i aug6 i

Tonic represents 'at rest', dominant represents 'pushing towards tonic', and subdominant represents 'moving away from tonic'. 'Subdominant' is as far away from tonic, in the opposite direction from dominant, as dominant is from tonic.

So, major or minor, IV (or iv) isn't dominant. It certainly isn't tonic. It has the tonic note in its formula, so it won't be pushing towards something that's already there.

It does, however, contain the subdominant note, which when included with the dominant triad, makes 'dominant' even 'more dominant'. It also contains ♭6 - which, in another guise, as ♯5, could easily be part of V+, again, a dominant variant.

Lack of the leading note, though, obviates its role as anything but subdominant, major or minor, in major or minor key.

is '4 minor' sub-dominant?

Yes.

With no other info I would assume "4" means the Roman numeral in harmony IV, and when the chord is minor it's written in lowercase iv.

Whether the chord is major or minor doesn't matter, it's the root of the chord that makes its identity a subdominant.

'4 minor' can be substituted with ♭VII7 (backdoor V of 'backdoor ii V') which often resolves to 'Tonic'.

I think this is mixing up the basic elements of the backdoor progression. In that progression the ♭VII is a sort of substitute for V as both chords can harmonize tone ^2 and ^4 of the scale. ♭VII and V sort server the same function in that sense as dominants.

The subdominant comes into the backdoor progression as a sequence of roots by descending fifths. If ii V I is descending fifths to the tonic, the backdoor progression maintains part of the root changes by preceding the ♭VII with a descending fifth from the subdominant IV. The minor iv is used further the borrowing from the minor mode.

Interestingly, both progressions end up with the same basic functional categories. Chords ii and IV are considered forms of subdominant, pre-dominant harmony. Again, ♭VII has a substitute dominant role. So IV iv ♭VII I and ii V I both fulfill the same function: subdominant > dominant > tonic. The backdoor progression gets the job done with a lot of modal "darkening."

'4 minor7' is temporarily non-diatonic and contains the ♯5 of the key centre as well as the ♭3.

Just one quibble. iv7 would be tones ^4 ♭^6 1 ♭^3, in C major it would be spelled F A♭ C E♭.

If the tone really was ♯5, it would be G♯ in C major. You might find that in a chord like E7 moving to Am. The raised tone then takes on the role of dominant harmony.

That brings up a point I think worth making. Tonal changes in the direction of the subdominant involve lowering tones, think adding flats. Tonal change in the direction of the dominant involve raising tones, think adding sharps.

Raising a tone for ♯5 suggests a leading tone and dominant harmony whereas lowering a tone for ♭^6 suggested borrowed harmony.

It's interesting how all the lowered tones of the backdoor progression both use borrowed chords and put an emphasis on the subdominant.