Hi Stuart, thanks for posting the pics....
What I propose to do here is tackle your problems in 2 parts.
The first part will be to get the prob with clutch and cylinder reel adjustment and chain tension corrected. To do this (and considering the good condition of the sprockets,) I believe the reel bearings may be shot, causing the snatching/grabbing you describe
The second part will be to hand you over to Grumpy (our Briggs guru) to get the engine sorted and running sweetly.
What I need you to do is:
(1) Remove the cylinder reel following the instructions contained
HERE You don't have to remove the soleplate, we just want to make sure the reel bearings are serviceable. If in any doubt....replace them....(I would anyway, so I then know their history

)
(2) Re-install the reel and adjust the cylinder reel to bedknife (bottom blade) clearance correctly. You will find the instructions for doing this is contained
HERE Once that has been completed, can you please let us know and we will proceed on to the clutch adjustment.

Just be patient and proceed carefully...any questions, please post to this thread. Best of luck!