Hello, and welcome!
1. Most maxi coolers have a pre-set temperature, around freezing (though some use a food-safe form of glycol to allow them to go colder). They can be bought with different lengths of product (beer) coil as a crude form of controlling their cooling effect or to a degree the rate at which beer is run through when serving can affect it. Though one can DIY alterations if required.
2. I can't help with specifics, as haven't bought a regulator for a while. But make sure it is a beer regulator with adjustable pressure - some are designed to control flow rate rather than pressure (e.g. for welding) so have the wrong sort of gauges and some may have no pressure relief valves for safety. Particularly important if buying beer in containers without built in PRVs.
3. You can get posh towers (e.g.
this style) right down to cheap plastic dalex taps or even party/portable taps (e.g.
this) that you hold by hand. As you can see, a huge range in budget possibilities, sometimes they come up used on ebay too. Some come with flow control adjusters, which can help fine-tune the dispense rate.
4. 3/8" is fine for gas line unless you need it to be very flexible. For short runs with beer line then yes 3/16" is much better even though adaptors may be needed. It has a tiny bore which heavily restricts the flow, causing pressure to drop significantly along its length whilst dispensing, a couple/few meters will often be sufficient. Attempting to balance the line with 3/8" instead might require dozens of meters for a similar effect. The reason for balancing the line restriction to the keg pressure is partly to avoid excessive flow rate and partly to gradually reduce pressure towards the tap, excesses of either can cause foaming.
5. The recirc lines are indeed for water, not beer. They are designed to recirculate water through lines bundled with beer lines within insulation (python lines) and so keep the beer from warming up in the line. Aside from the beer being warmer in the glass, warmer beer doesn't retain CO2 as well, so this can try to come out, causing foam in the lines. Though you may not need this if the beer lines are short or the weather not too hot.
Hope thats a useful overview anyway,
cheers
