1. What are the best choices of water filter systems (water purification?) for residents living in locations where they cannot easily harvest rain water from their roofs, like those living in apartment blocks or for those who only have access to tap water?
On the cheap end, you can put an inline canister filter (or multiple canister filters in series) that works just based off household water pressure. These can filters can accept woven fiber filters (for suspended solids like clay, silt, bugs, etc.), ceramic filters (for much smaller suspended solids like bacteria and viruses), activated carbon tubes (for organics, residual chlorine, etc.), and you can even get demineralization resin for them (to remove specific ions like sulfates or magnesium, etc.). You can get really complicated with blown polyethylene cartridge filters followed by ultrafiltration then two-pass high-efficiency reverse osmosis with a reject recovery system and post-permeate salt dosing for flavor, but that's going to cost you. You have to ask yourself, what's my budget, what am I worried about being in my water? If you're in the US and you have a city water source, I wouldn't worry about anything. If you live on top of fracked ground and you have well water, I'd sue them and put in something really nice like the second option I described.
2. What are the best choices for removing chemicals (Flouride, chlorine etc), metals and removing harmful biologicals?
An activated carbon filter is about the best catch-all out there for most organics and for oxygen-demanding chemicals, and they're cheap. If you're worried about fluoride, radioisotopes, and heavy metals, a bone char filter is one of the most effective filters, but bone char can be expensive. You could make your own (when you make charcoal, throw some cleaned bones in there). RO will also remove these contaminants, but at home you'll lose a lot of the water you're purifying this way.
3. What are the most sustainable and efficient water filter systems?
Any gravity filtration (multimedia, carbon, sand, etc.) is going to be more efficient in terms of energy than an induced pressure (RO, UF, NF, etc.) filtration that requires a pump. Most forms of filtration are "dead-end" processes - for every gallon of water that enters the filter, you get a gallon of clean water out. Some (like RO, as others have noted), have both a clean water (permeate or filtrate) stream as well as a waste water (reject or concentrate) stream. In some filters, a portion of this waste water may be recycled to the beginning of your filtration system and purified again to increase the recovery (percent water recovered from filtration).
4. What are the best ways to test water quality and how frequently and how thorough (or costly) should the test be? ie what would be the best weekly/monthly test?
You can go as basic as a pool test kit (for pH, chlorine, and hardness) with periodic lab screening for biologicals. You could go crazy and get the equipment to test TSS, TDS, BOD, COD, and biological testing at home (it's not *that* expensive, maybe a thousand dollars or so?).
