Is this a well constructed roof

Since you're putting a gable roof on it, I can only assume this is being built for storage purposes, not for aesthetics.

Building it with a glulam ridge beam would be more difficult, but it would be much stronger. The main issue with a ridge beam design would be lifting the beam into position. If you can manage that pretty easily, the rest would be a breeze. I'd put some ceiling joists up at either end, lift one end of the beam over one of the joists, pushing it far enough to clear the ceiling joist on the other side, like so...

B
   B
      B                 
  JttJttJBttttttttttttttJttJttJ     
  S         B                 S        
  S            B              S
  S               B           S
  bbbbbbbbbbbbbbbbbbbBbbbbbbbbb

B = ridge beam
J = joists at the ends of the barn
S = studs at the ends of the barn
t & b = top and bottom plate

... and then lift the bottom end of the beam up with some ropes. The joists will need some strength to support the ridge beam, and you'll need to be able to stand on them, so I'd use three 2x6's @ 9" OC on either end (where the outermost "2x6" is actually the top-plate of the walls on either end of the diagram) with some 5/8ths plywood on top of them. I'd toss in some temporary studs at the center point to make sure the joists hold up to the weight of two people and the ridge beam. Once the beam is in place, put in the ceiling joists and plywood on top of them to make walking around easier (and to create an attic for extra storage,) then put the rafters up.

Grab some metal brackets for the joists, rafter heels, and rafter/ridge connections (it'll cost you an extra $125, but it will make the structure a lot stronger) from a HI store and your little barn thing will be able to withstand some nasty winds.

Cold and hot roofs are an unfortunate terminology that leads to misconceptions. I shouldn't complain, since I'm not sure I have a better idea. Perhaps ventilated or unventilated sheathing? That is the deciding factor.

In cold roofs, the outside air is allowed to freely flow under the roof sheathing. In hot roofs, the insulation is typically installed close to the roof sheathing, but the main characteristic is the space under the sheathing is closed to fresh air flow. Thus, your unventilated roof is indeed a hot roof. You should be aware that according to most building codes, all structural voids not filled with some material should be ventilated.

The main advantage of cold roofs is for cold climates. Snow is much less likely to melt, and if it does melt, ice dams are less likely to form, or at least will be much smaller. With unventilated roofs, heat leaking from the building melts the snow. The water flows down to the eaves, which are much colder, and freezes. This causes a tiny dam which inhibits the water flow, which causes more water to freeze, making a bigger dam. The cycle repeats, and many tons of solid ice can accumulate.

Not only is the weight on an overhang an issue, but the water that hasn't frozen yet just pools. Most sloped roofs are not designed to resist pooling water. It easily backs up under the shingles and other joints and leaks into the building. This is particularly a problem in valleys.

In warm climates, a cold roof is still an advantage, but much less critical. The ventilation prevents any significant temperature differential from building up in the roofing system. The smaller the temperature change, the longer the roof will last. The ventilation will also carry off any moisture that may enter the space, avoiding mold issues. Unventilated roofs must have a good reliable vapor barrier, because any moisture entering the space will have nowhere to go.

The only advantage I can think of for a hot unventilated roof is it allows cheap, easy construction of vaulted ceilings without any special details to ensure the sheathing is being ventilated. These special details often end being a double sheathing system which significantly adds to the cost of a vaulted cold roof system.

You can use foam insulation of any type with either roof system. It's all a matter of correctly detailing the roof system to ensure it is either ventilated or that all voids are closed and filled.