Speed limit - 50 Single carriage way, 60 dual carriage way. End of. ( a motorway is just a special dual carriage way)
Insurance .. your responsibility at all times. Hire company will not do it automaticaly but may arrange for a fee. Most fully comp insurance covers you whilst towing for 3rd party losses.
To check electrics go to a cravan servvice centre .. most of them have a plug in check board ... you may be able to blag a free check
As said ... unless you have "grandfather" rights or have taken a towing test you can only tow up to 750 Kgs .. and that includes the weight of the trailer. If you have BE on your licence you are OK .. if you don't .. its 750 Kg.
A car trailer I assume is to collect a car ... check the weights properly .. even if you have a BE licence it is illegal to tow above the max permitted weight for the car .. check the VIN plate, then check your trailer weight when loaded. If you exceed either max trailer weight OR the max train weight (car + loaded trailer) your are breaking the law and probably the conditions of insurance as well ..

HTH