All driving licenses cover category B. This allows you to drive a vehicle of up to 3500kg with a trailer with an MAM of up to 750kg or a heavier trailer provided the sum of the vehicle MAM and trailer MAM does not exceed 3500kg and the trailer MAM is less than the unladen weight of the vehicle. B+E is only required if you want to use a trailer with an MAM of over 750kg and the sum of the vehicle MAM and trailer MAM will exceed 3500kg. If you have B+E can drive a vehicle of up to 3500kg MAM with a trailer.
If you took your driving test before January 1st 1997 you will almost certainly have B+E (and you will actually be subject to different, more generous, rules on weights). After that date you will only have B and need to pass an additional test to get E. Anyone passing the additional test after 19th January 2013 is limited to a trailer with an MAM of no more than 3500kg. I am assuming that you only have B and that you passed your test in 1997 or later.
Ifor Williams Trailers Ltd say that B+E is required because, if you are towing a trailer with an MAM of 2000kg, the unladen weight of the car must be at least 2000kg, so its MAM will be more than that. The vehicle and trailer combination will therefore exceed 3500kg MAM, so category E is required.
The 1.6 Ecoboost has a Gross Vehicle Weight (another name for MAM) of 2045kg. So no, you cannot use a trailer with an MAM of 1500kg. The maximum permitted is 1455kg. But you can use a 1455kg braked trailer provided its loaded weight is no more than 1000kg. If it is not braked, its loaded weight must not exceed 750kg.
To answer your last question, you cannot use an unbraked trailer with an MAM of 1500kg. However, if you used an unbraked trailer with an MAM of 1455kg or less and the unladen weight of the trailer is 357kg, you can put up to 393kg stock and equipment in that trailer, taking the total weight to 750kg. I would recommend keeping well away from that limit to allow for any errors in weighing.
Given that your vehicle is limited to 750kg unbraked towing weight, I'm not sure why you would want to go for a trailer with a higher MAM. It would be simpler to get one with an MAM of 750kg. That way you are clearly legal unless you overload the trailer.