-how do you play the first bar? what time is this? 4/4? (we count 12 eighths!)
It's traditional not to bother writing 3 for the triplets.
I fully agree with the first part of this answer concerning the rhythm performance and note values, also the history of notating. So it's a little kind of nitpicking to defend whether a 12/16 bar is actually a 4/8 with triplets or not as I did in the other question (s. link above) respectively 4/4 with triplets or 12/8 in these Gigue here. But it is certainly not wrong if we discuss the time of a Gigue.
-which hand is playing which notes?
we have 2 videos: Richter and Sokolov. They play both the notes stem up with the left hand. replete as an organist plays them with the right hand and the triplets with the left hand. Obvious both solutions are possible.
When Bach wrote it in purpose to do so he could have written ad libitum.
-how will you play the second and the following bars?
This was the easier part of the question: "Simile" means similar!
in the same way (continuning).
With other words: when we study a new piece by Bach - or a similar piece of his period with similar pattern we can also reduce any piece by Bach with a "simile" pattern by notating or memorizing just the chords like Bach did here. I wonder why he didn't so in his prelude in C (WTC 846)