WARNING - I AM A BEGINNER, but I have solved the runaway chiming on an identical clock.
My Perivale Bentima had some similar issues. In particular the runaway chiming that occurs at about seven minutes before the quarter chimes (on each quarter, not just the first one).
On my clock which is identical to this one, I found that when the clock went into warning, the locking detent was ineffective and allowed the chimes to run on and on until the minute hand's movement allowed it to stop. To my inexpert eye, holding the chimes back and allowing them to operate correctly requires two stop devices to operate; one on the count wheel, and one on another wheel on the backside of the front clock plate, behind and slightly above the count wheel. In my case, these seemed to be slightly mistimed, in any case, I solved that problem by changing the timing very slightly.
The notched wheel on the back of the clock plate which engages with the locking lever (it may be called the 'locking detent') rotates at about three times the rate of the count wheel. This made it a bit tricky to see where exactly the wheel needed to be in relation to the count wheel, but you can stop the chiming mechanism quite easily and see that when the locking lever has dropped into the notch on the count wheel, stopping the chimes, it has also settled into the rather longer notch on the wheel on the other side of the clock plate. If these are mistimed, like mine, you will see that the locking lever has not FULLY descended into the notch of the count wheel. It can't do so unless the lever is also fully in the notch on the detent on the other side of the clock plate inside the mechanism. Fortunately it is easy to see this by looking into the top of the chiming mechanism.
There are two grub screws on the rear of the count wheel which allow it to rotate on the arbor, and IF NEEDED, the timing can be altered by loosening these and slightly moving the arrangement of the count wheel on the arbor and hence in relation to the rest of the mechanism. Don't do this unless, like mine, your locking lever is plainly not sitting in the detent notch on this inside wheel while on the front of the clock the lever is lined up with the notch in the count wheel.
In my case a small change was all that was needed to make the runaway chiming stop.
I won't comment on the strike part of your problem except to ask if the strikes you did get, were initiated by the clocks normal operation (albeit speeded up by turning the hour hand) or were set off by manually lifting the flirt which is also the locking lever on the chimes. My clock would strike whenever I lifted that lever marked as 'locking lever' in the photo up above, but they would not happen when the clock's own mechanism was lifting the rack hook. It almost set it off, but was about 25 thou of an inch short of lifting it high enough. I solved that by adding a smidgen of solder to the top of the strike flirt where it engages on the rack hook and triggers the chimes.
If you can lift that locking lever up manually to its full height and the chime mechanism doesn't operate, then your Bentima has a different problem to mine.
I found this mechanism VERY sensitive to the smallest adjustments or handling. Even putting it back in its case face down on a cushion was enough to make it work or stop working because gravity altered the disposition of the locking lever, moving it forward or not when it was face down. I am glad to say that it is now working fine, but it took some time to get it there.
ONCE AGAIN I STRESS THAT I AM A TOTAL BEGINNER AND YOU SHOULD KEEP THAT IN MIND.
This video shows my Bentima going through its chiming cycle after I had made the alteration mentioned above. Keen viewers will spot that the strike mechanism isn't operating properly since the strike flirt is not lifting the rack hook high enough to drop the rack right down to its proper level in relation to the snail on the hour arbor. This was taken just before the 'solder trick' which solved that problem.
VIDEO:
http://www.youtube.com/watch?v=vTUCHnF-j_E