So I have been doing the 4 note ascending/descending sequences (1234 2345) of G major at fret 3. I noticed that after a while the muscle below the pinky (digiti minimi according to the picture...) starts to cramp/strain.

I know I should feel relaxed when playing. But how do I know if this strain is coming from me tensing up or the lack of strength and endurance of that muscle? Does this strain go away as the strength and endurance is improved?

You might not be entirely ready for that kind of stretch yet; try doing the same pattern higher up on the neck and get 100% comfortable with it there then move it down a fret at a time, making sure you're comfortable with it at every position. If at any point you're not comfortable, move back up a fret and make absolutely sure you're good in terms of the feeling in your hand and your technique then try again.

It might very well take quite a while to get down to where you want to be but making sure you're ready for it is more than worth it; injuring your hands with unnecessary stretching is obviously terrible.
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I cannot be sure, without looking at your technique and seeing you play, but that muscle is used in bringing your little finger round as if to touch your thumb.

My first suggestion would be- as has been said- that you're not used to the stretch. However, make sure your thumb isn't too high on the back of the neck (which would cause you to need to move your pinky finger round like that to reach the fretboard).

The further up (towards the sky) you have your thumb, the further away your fourth finger will be from the fretboard, generally. You might be straining for this finger to reach.
The answer is almost always over-exertion. Your fingers are already more than strong enough to push a little metal wire 1/16 of an inch, and have the endurance to use a pen for thousands of words on end.

Technique is about focusing that physical exertion. Using poor angles and inefficient movements means you have to put more muscular effort into playing than you need to.

Bone up on basic technique and do your exercises slowly with a metronome. Find whatever extremely slow tempo gives you complete control of the sound of every note, and start there.
Tip my guitar teacher told me.

Lay your hand flat on a table or something and stretch your fingers as far apart as you can. Then take other hand and stretch a little further. Do this every day and you should get more range from your hand. I actually got about another 2 inches