BoxCar

That's a boat load of questions and right on point. As far as antennas, height is king. The higher the antenna, the further it is to the radio's horizon and coverage area. The type of antenna is dependent on what you are attempting to accomplish, the height above the average ground level in your desired coverage area and what the area contains as ground cover. Ground cover includes buildings so include them in your calculations.
As for building out a repeater, go to repeaterbuilder.com for examples of needed materials and issues with the planning and construction of a repeater. For your power supply needs, add the required amperage needed for the transmitter running at its highest power and the amps needed for the receiver (and yes - you do need both) then add 20% for overhead.
The questions you haven't asked will be about gain and losses in your transmission line and antenna. Antenna gain is reported using 2 scales, dBi and dBm. The one that counts is dBm as this is measured using real power rather than theoretical Dbm is dBi minus 2.l or a difference of about 40%. A 3 dB difference is either 50% less or twice the input power measured at the antenna input depending on if it is loss or gain measured in watts. For the best estimate of how your system operates, either as a base station or repeater requires a VNA or Vector Network Analyzer as it's the best device to measure transmission loss in your feedline and to tune your antenna to resonance. Yes, all antennas require tuning for optimal performance. 
Good luck on your dream.