Yes, the spare tire size would definitely be the same size as the 4 on the ground. The 205/75-R15 size sounds right if you have 15" wheels. There may be a code stamped on the wheels you have that gives you the correct wheel lug spacing to match yours. If not, a tire shop such as Discount Tire can help you. They are a good source for trailer tires and wheels.
Load Range D is good if you can get it. I know Maxxis makes Load Range D in that tire size. I'm sure your trailer came with Load Range C tires as OEM equipment.
Hi-Lo 1707T - Tire Minder TPMS on Tow Vehicle and Trailer, 300W Solar Battery Charger, Equal-i-zer WDH
2014 F150 Platinum 4x4 3.5L EcoBoost SCrew