I drive a Nissan Leaf, though DW has a big Peugeot diesel estate which is the 'family' car for when we need to do long distances or carry lots of stuff.
To address your specific questions, EVs don't 'start' as such. You press the button or turn the key and they are ready to go. The electric motor only runs when you are actually using it to move. In extreme cold climates the battery performance will be substantially degraded, but it rarely if ever gets that cold in the UK. In those markets Nissan fit a battery warmer. Additionally, most pure electrics can pre-heat from the mains. Otherwise, cold weather reduces the battery performance and hence the range - whether that matters depends on the journeys you are doing. Expect to use more electricity in winter than in summer, but the same is true for petrol and diesel cars; it's just perhaps less noticeable.
There isn't a system for booking charging points, unfortunately, but at the moment as long as you can be a bit flexible and wait if necessary the supply seems to be adequate compared to the demand, at least in the area of the country in which I hang out.
Battery lease was/is essentially a Renault thing. Most manufacturers don't do it. Nissan, who are closely tied to Renault, did try battery leasing but it really didn't work. The theory was that people would pay for the security that the battery would be replaced if it degraded too much, but all the evidence is that batteries are lasting better than feared. Personally, I wouldn't lease a battery on its own. However, a lot of people seem to use PCP deals which include the battery.
There is actually a lot less servicing to do on a 'pure' EV - there's no oil, oil filter or air filter to change, no spark plugs, no engine radiator and coolant, no exhaust, cat, DPF &c. Brakes tend to last a long time since most braking is done by turning the motor into a generator and recharging the battery, with classic brakes only being used for the final stop, when the battery is already full and emergency braking. Nissan do a fixed price scheme at their dealers but I haven't had mine long enough to have it serviced.
The go-to place is www.speakev.com which is the main EV owners site for the UK and deals with general issues and vehicle specific ones. The people there are generally quite friendly to those thinking about taking the plunge. There are also various reviews of the available EVs, buying guides and the like to be found on YouTube.