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Fullback gets a bat and can be sin binned LBW for catching a ball that is going to hit the padding on the post
You are right, there doesn't seem to be any IRB sanctioned set of Tens laws. From what I've seen though, that is probably because they would be pretty limited at the top level - there wouldn't need to be much difference other than length of the match, mandated numbers at the scrum and maybe modifications to penalty taking.
Having a look around, this gives an overview of the RFU's recommendations for all the 15s variants and probably isn't far off the mark. The key to all of it is probably the opening statement that the intention is to "slow down the pace of the game to incorporate lower levels of fitness [and] control the levels of contact to allow for different levels of ability". When those aren't an issue, the laws and regulations don't need much change.
For mine, I've always seen the difference in popularity being one of differentiation i.e Sevens is more identifiably different from 15s and more accessible to the average player - as Beige noted, perhaps a bit easier to come at from a touch background. By comparison, Tens still has all the various aspects of forward play and once you have that it is a short hop to just playing 15s. A bit like 50 over cricket, it perhaps falls between two stools...too abbreviated for the purists, and not as flashy as the even shorter form. Sevens has also been very successfully branded as the festival game, with much of the interest being in the event as much as the play. But I personally like Tens as a short form and certainly see it as an under-utilised tool in player development, particularly if Sevens ends up getting hijacked by Olympic status and becomes an end in itself. The last thing I would want to see is promising young players starting in 7s and then never coming back. Giteau, O'Connor, Cummins - they have all been through the 7s path and if the day came when they could build a long and lucrative international career in that form of the game....?