A few thoughts…and I speak as a CPC-Holder here (Transport Manager’s qualification) as well as a driver, so hopefully I can give you a fairly definative answer… 
- As has already been said, with a provisional LGV cat C entitlement you will need a full LGV cat C licence holder to sit you up in order to be legal on the public road. That person must have held the full LGV category for a minimum of 3 years. You must also have insurance to cover a provisionally licenced driver.
Yes, you would need to use L-Plates. You also need to remember that a provisional entitlement is only valid for 2 years, so you would need to pass your various tests within this time to avoid having to pay to have it re-added to your license.
The good news is that, yes, you can drive on motorways as a learner. An LGV Test is effectively an advanced driving test, therefore the law assumes that you are basically capable of driving in the first place!!! Some test centres - depending on where they are - actually include a short motorway stretch in their test routes. 
- Provided that neither the vehicle or activity for which it is used earn you any money (prize money excepted
) then you will not need an O-Licence.
If you earn any part of your living from the horses, but are only transporting your own/those of your friends, then you will need a “Restricted” O-licence.
If you transport horses for “Hire or Reward” the you need a full O-licence.
I won’t go into the differences between the 2 types of licence here, since you say that the lorry will be used purely for pleasure, therefore you will fall into the first category. 
As for tachos, again, provided you don’t need an O-licence then you needn’t use one. However, if the vehicle has one fitted then the speedo won’t work without a “card” in, so you’ll need to insert one for that reason alone, even if you only change it when it gets so tatty that it starts to jam the instrument up. 
You could, of course, just have the tachograph removed and replaced with a basic speedo (although confirming the legality of this from a plating point of view with your local LGV Test Centre would be wise), but personally, I’d leave it in - if only because you never know what you, or a future owner, may want to do with the vehicle in the future…with the impending law-changes regarding Digital Tachos, refitting with new equipment at a later point may prove costly, and it’d be a shame to limit your options or reduce potential resale value for the sake of bunging in a fresh bit of card from time to time. 
Obviously, as soon as the vehicle comes under any type of O-licence, the driver also becomes subject to Drivers Hours Law.
- As you quite rightly point out, “downplating” to 7.5t would be foolish because of the reduction in payload that would result…another pitfall which many ill-informed horsebox owners have discovered at great expense involves axle-weights. A fully-fitted living area designed and legal in a 10 tonner will often result on the vehicle being overweight on the front axle if it is downplated.

One other thought…as a 10-tonner, the lorry must have a working speed-limiter, set and sealed to 60mph*, unless it is pre-E-reg.
If it doesn’t have one, you will probably have to have one fitted in order for it to pass an MOT…and if it does, it must work. That’s the case regardless of whether the vehicle is being used privately or otherwise.
Quite why anyone would want to hurtle 4 valuable horses and their nearest and dearest down the road at more than 60mph is beyond me, but there are always a few who try and avoid this one. And fail. 
There’s a guy called Krankee who’s an ex-Traffic Officer and hopefully will be able to confirm all this for you…if he doesn’t appear on the thread, a PM or a post in the Safety and Law Forum will get his attention. 
Hope that helps…
Ps. Before the nit-pickers steam in, yes, the limiter needs only be set to 60mph at the moment…it is over 12t gross that the legal setting is 56mph. This anomaly will AFAIK be ironed out with the new legislation bringing 7.5 tonners under limiter law later in the year, so check to be sure. 