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This is in key of C.
For the first three notes in the Bass Clef with LH, I want to do middle, index and thumb.
But afterwards I run out of fingers for the next half note, is it acceptable for me to use thumb again? Or move whole hand to the right and use pinky?
Just going by the notes in the image, I would not repeat the thumb or cross over, but instead treat the fingering of the first part (G A C E) like a vi4/2 chord that is arpeggiated, then exchange fingers 1 and 5 silently on th E3...
...then the jump up to E4 is between fingers 1 and 5 which should be the familiar distance of an octave.
Later change fingers for the repeated E3.
Don't ever jump with your thumb.
In this case if you have to use the middle finger on the G, just roll over your thumb while its on the C and hit the E with your middle finger. This aviods another thumb jump going to the next E as well.
If you end up on the low E with your thumb, try transferring the note to your pinky without re-sounding it so you can smoothly play the octave if the music style requires it.
Beyond the basic requirement to play the notes, the purpose of fingering is articulation.
Most musical instruments have some "bad habits" which trap beginners into poor playing. On keyboards, the worst habit is the idea that successive notes should always be perfectly legato.
There isn't really any context in the OP's example to show what "should be" legato and what shouldn't, but if there is no good reason why the C and E should be legato, by all means play both with your thumb if you want.
If they should be legato, then of course you need to find a different fingering.
The same applies to the octave jump between the two E's. On piano, the sound of the long E will inevitably decay, and nobody will notice a short gap at the end of the note. Swapping fingers from 1 to 5 to play the octave legato is a pointless complication. If the jump happened to be a bit bigger than an octave, then what would you do?
If you really want to sustain the long E for its full length, pianos have a nice feature call the sustain pedal. Just press it after you play the fourth 8th-note, and you don't need to use your finger to hold down the long note, whatever fingering you use.
Or as another alternative, play the 16th-note E with your right hand.
It's very rare there is only one way to skin a cat.
