IMHO, the cock needs to be replaced. Look at the enlarged photo in the first set and it can be seen that the bottom jaw is too short to properly hold a flint, especially with a thick leather wrap, as in the top photo of the second set.
IMHO, the top jaw screw is set too close to the front of the jaws, restricting how solidly the flint will be held in the jaws.
Using a thinner leather wrap, similar to the one in the first set of photos, or cutting an oval out of the rear of the thick wrap, to allow the flint to set farther back, might help. However, the rather thick top jaw screw, in combination with the distance from the back of the cock extension, will continue to cause problems until it is fixed right.
IMHO, the screw holes can be filled and a new top jaw screw, of the proper diameter, can be relocated, or the cock replaced.
IMHO, it would be easier to replace the cock.
As to the issue of the number of flashes in the pan, some of these cheap pieces had a weird flash channel that made a turn inside the breech. Another problem with this type flash channel is that it's too small, all the way through.
An acquaintance had a cheap, un-named piece, with a similar lock. Once we got the lock working, the small diameter flash channel, combined with the turn, made ignition rather...erratic.
Even with enlarging the diameter of the outer flash channel, the small diameter of the inner flash channel, after the turn, caused problems.
I don't know if this piece is made with the off the wall flash channel, but it might be something to check.
God bless