Post-Transformer RMS Voltage for This Circuit

You can make some simplifying assumptions. They will add less noise to the result than the fluctuation of line voltage this circuit will encounter, so don't worry about it too much.

Assume that the capacitor is charged at the peak of each line cycle, and then discharges thru the resistor the rest of the time. The cap therefore has to not drop more than 2 V during 17 ms. You say your acceptable output voltage is 14 to 16 volts, so assume the cap gets charged to 16 volts. Discharging from 16 to 14 volts would take .134 time constants, since .134 time constants is 17 ms, you know that a full time constant is 125 ms, which is R x C. 125 ms / 150 Ω = 830 µF.

Of course in practise you don't want to cut it that tight. At the very least, you want a 1 mF capacitor, but I'd probably use 1.5 mF or 2 mF. That will be electrolytic at this capacitance and voltage level. It should be rated for at least 20 V, although 25 V would provide longer life.

Note that the current provided by the transformer will come is large and short spikes, and their level will be directly proportional to the input line voltage. For robustness, you should design this circuit to operate properly from at least 105 V to 125 V AC. This gets tricky. There is a reason we use switchers and regulation nowadays.

Your transformer has 0.35A rating for its secondaries (each). You can fuse those with a [say] 0.3A fuse each. (The center tap doesn't nee a fuse.) The fuse in the primary [if you want it] needs to be much smaller. For 115V primary, you have a turns ratio of about 3.20 (to both secondaries in series), so the max current in the primary is about 0.11A. A 100mA fuse should work there.

Beware that if put the secondary side fuses after the rectifier they need to be DC fuses. AC and DC fuses are not the same because the latter need to cut the arc while in the former it gets cut by the AC sinewave itself.