There are only two types of trike that you really need to know about, bike engined and car engined. Having said that, it isn't the engine type that finally influences the decision of what type of trike you're going to ride, it's the weight of the trike, and what you are actually licensed to ride; any type of trike can be built or bought to suit disabled needs. These can realistically vary from a humble 250cc bike engined trike, all the way up to a growling, car engined V12 monster.
No, neither for bike engined nor car engined trikes. Helmets are only compulsory for motor bicycles, not motor tricycles. On the other hand, you may wish to wear a helmet, say on a long journey, to protect your head from the weather, flying debris etc, etc; it's up to you. As a word of warning, should you be unfortunate enough to have an accident on your trike, some insurance companies may try to use against you, the fact that you weren't wearing a helmet should you wish to make a claim for damages, especially if you sustained head injuries.
Seatbelts on trikes
Vehicles first used before 1 April 1987.
Seatbelts are required for three wheelers that fall into the following categories:
1. With an unladen weight over 410Kg first used on or after 1 January 1965.
2. With an unladen weight over 255kg if first used on or after 1 September 1970.
Less than 410kg unladen equipped with a driving seat of a type that requires the driver to sit astride it and where that vehicle was constructed or assembled by a person not ordinarily engaged in the trade or business of constructing those sort of vehicles.
Vehicle first used before 1 April 1981:-
Drivers - A belt that restrains the upper body(but need not include a lapbelt).
Forward facing rear seats - no requirement.
Vehicle first used after 31 March 1981:-
Drivers - a 3 point (lap/diagonal) belt. 
Forward facing rear seats - no requirements.
Vehicles first used after 31 March 1987:-
Seatbelts are not required for trikes that either ...
1. Have an unladen weight of less than 255kg.
2. With an unladen weight of more than 255kg but less than 410kg which are equipped with a drivers seat requiring the driver to sit astride it and where its been constructed or assembled by a person not ordinarily engaged in the trade or business of manufacturing vehicles of that type.