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When one is playing a 4/4 tune "finger picking style"
(Thumb base note, Index 3 string, Middle 2nd string and ring finger 1st string) (Not coincidentally 4 "actions/picks")
My question is:
Would a bar be the above action ONCE or FOUR times?

If the time signature is 4/4, there are always 4 beats in every bar. But that doesn't have necessarily anything to do with how many "actions" you perform per bar, i.e. how many times you pick/finger a string or anything.

How many "actions per bar" you do depends on the rhythm pattern and how you realize it, and there are any number of different rhythms and ways to play them.

Here are a few example patterns. The first one is a boogie rhythm with a fast tempo, 200 beats per minute, and the guitar comping rhythm pattern has two rhythmic "actions" per every beat. In other words, it's a 1/8th note pulse, with swing. On every beat, there's a bass note, and the rest of the chord notes are fingered in between every beat. It's a fun little rhythm.

The next one is some kind of a ballad, with a very different feel between the bass notes and the higher notes. Again, the pulse is 1/8th notes with swing, but the tempo is much slower, 90 beats per minute.

The third one has the slowest beats-per-minute tempo, 80 bpm, but the guitar comping pulse is 1/16th notes (with swing). Even though there are fewer beats, there's more actions to do for the guitarist.

(Oops. All of them are missing a time signature marking. Oh well.)

It could be. And it could be doubled to play twice in a bar, the second time with a 5th of the chord instead of the 'base' note. It depends on what the song sounds better with, and how you want to play it anyway. It could be played as thumb on beat 1, 3 fingers simultaneously on beat 2, then repeated for the 2nd half of the bar. Don't get too hung up on 'correct fingering', as mentioned in my previous answer.

It's up to you!

If you're playing some song fingerpicking style, then you're the one making up the fingerpicked part anyway. Therefore, you can do it as fast or as slow as you want. You could play the whole thing twice per bar, once per bar, four times per bar, et cetera. If you're the one making up your part, why worry about how it's "supposed" to be played?

What you are describing is one of many fingering patterns. Numbering the fingers t,i,m,r like 1,2,3,4 your pattern will be 1,3,2,4. You can construct other patterns like 1,2,3,4 or 1,4,3,2 or any other sequence or combination.

Now your question:how many times this pattern can be played in a 4/4 bar?

This depends of the speed you want to play the pattern: 4/4 can be divided in 8/8 or even 16/16 (and even shorter units). When you play crotchets you play the pattern once, 8th notes twice (as 8=2x4) 1 etc.