My first bit of advice would be to decide right at the outset how far down the realism rabbit hole you want to go. If your fantasy world is nothing more than a pretty backdrop to your story, the details really don't matter, realism (within reason) doesn't matter, and you should just go with what you want to go with.
At the opposite end of the spectrum you can meticulously create a fully realised world in intricate detail.
I personally tend towards the more realistic side of things, so I've put pretty extensive work into these sorts of things. I'd never do it any other way, but it's worth pointing out that going down that route means a LOT of extra work. And I put "lot" in capitals for a reason. Seriously, you cannot underestimate how much work it is.
To answer your specific questions above though:
The first thing is to understand the structure of feudal medieval living, and the different classes of habitation, because they're quite distinctly different. Towns and cities are not the same thing, at all.
The first thing is that both towns and cities are significantly smaller than what you or I think of as towns or cities, and the second thing is that villages are much larger than you or I think of them.
Medieval society existed in dense networks of villages, about 2-3 miles apart, in every direction. The villages only stopped when they encountered land that couldn't be cultivates. Villages were quite large, typically around 700-1,000 people.
Towns appeared at random, in the midst of these dense village networks. The primary purpose of the town was to serve the immediate area; it was a central trading centre for the villagers to buy produce that they couldn't make themselves, and to sell excess grain/wool/fruit/whatever their village produced. As such, towns were quite small, and there were a lot of them. A typical medieval town will only be between 1,000 and 8,000 people.
Cities are a different beast all together. While towns serve regional trade, and emerge based on local requirements, cities serve inter-regional trade, and emerge based on the routes that feed goods into the towns. Typical sites for cities include points where routes to multiple towns meet, and ports and harbours that bring in goods from overseas.
The key thing about cities is that they grow from towns, and originally would have functioned to serve only the local territory. So your best bet is to establish towns first to serve local villages, and then determine your major trade routes (which will be based on what the region produces, what it needs from elsewhere, and where good travel in and out of the area) and that will tell you which towns would become cities.
Your travel distance is off because primary traders - that is farmers and so on who sell produce they make themselves - don't travel between towns, they travel from their village to town. The distance you've calculated is probably about right for the area that a town would service, but that makes the probable distance between towns twice as far (35 miles), which is much more believable (actually it'd be more like 37 1/2 miles because you've got to factor in the 2 1/2 miles between the furthest villages for each town).
The distance between cities is going to be much greater, as cities only appear at major convergence points. Secondary traders (i.e. merchants) don't need to return home each day to the farm, but rather their sole job is transporting good between places (in fact they'd probably mostly hire cartage companies to do the actual transporting), and the goods they're trading may embark on journeys that last months or even years from where they're produced to their end user. The merchants will tend to live and have offices in cities, with their product dispatched out to various towns.
As far as climate zones, again there's degrees of realism, but it's relatively easy to do the basics, which will already put you well ahead of 99% of fantasy worlds (including Tolkien's Middle-earth).
The single most important factor in determining climate zones is prevailing wind, which is dictated by the direction your earth spins. If you want an earth-like world you're best to keep the same planetary characteristics as earth and just change the landforms, as altering the orbit, size, axial tilt, and anything else like that has a pretty enormous impact on your world's viability for supporting life.
So if you know where your patch of land is on your planet, you can use the prevailing wind patterns on earth to determine climate patterns. If you want to get more detailed you can also determine your major ocean currents, although that's a bit more complex than prevailing wind.
Basically, when it comes to wind, there's a couple of key things to remember; the air collects water as it passes over the sea, and loses it (as rain) as it passes over land. Where the air is forced up over mountain ranges it loses water more quickly, while over flat land it tends to retain water for longer.
There are two basic directions when it comes to wind; windward, or upwind, and leeward, or downwind (upwind being towards the direction the wind is coming from, downwind being in the direction the wind is blowing to).
We can establish two basic characteristics of land; land on the windward side of a mountain range will be wetter, while land on the leeward side of a mountain range will be drier. The taller the mountain range, the more pronounced the difference. South America probably illustrates this better than any other place on earth, because of its clear linear shape and very tall central mountain range (the Andes). Take note of how the wet and dry sides of the range reverse at lower latitudes because the prevailing wind direction reverses.
If you want to get more complex, ocean currents play a part too, increasing or lowering local water temperatures which affects how easily the water is absorbed into the atmosphere. For example the Gulf Stream current makes the east coast of the USA much wetter than it otherwise would be.
Rivers, obviously, tend to form more on the windward side, where all the rain is going. They will flow always downhill, and as a rule rivers will join but not divide. Lakes form only where there's a recession in the land that a river or rivers drain into, but can't immediately drain out of. The basin will fill up until the water level finds a way out of the basin. Lakes only ever have a single outlet. (The best way to think of a lake is as a section of river that has briefly got very wide).
That's the basics, you can go into a lot of detail with all this if you wish.