Best use of Missionaries

1. Near or far? Should I spread the religion to a couple cities I just founded, that will probably get it anyway, to speed things up? Or should I send the missionary far away, to create a new beacon of truth?

First spread to your own cities, as you benefit from both the founder and follower bonus. You can make an exception here and spread it to a city state if they ask you to.

Try to spread nearby first. This creates a feedback loop of religious pressure which means your religion isn't as easily pushed out.

You could also use a missionary as a scout to make contact with other players, though I don't think their lack of combat defense makes this a valuable tactic.

2. Spread or concentrate? Use one charge in each city, or all three in one city?

Depends. Usually, even if a single spread does not convert the city, it tends to accelerate the process enough. It really depends on the size of your empire and whether you're looking for short-term conversions (= spam spread) or long-term nurturing of a wide religion (= spread the spread around).

3. Green fields or fierce competition? If using a far strategy, do I seed the capital of the competing Civs who are about to get religion, to try to crowd it out before it spreads, or the other Civs, to try to spread there while having no competition and stay clear of Inquisitors in the near term?

I suggest spreading to green fields first. Religious combat/competition will generally detract from the total religious output you can muster (as your opponent negates you and vice versa).
If, however, you first grow your religion on green fields, it will be much stronger to then take on other established religions. But then again, your opponent gets the same time to grown their religion. So I guess if you're ahead, try to crush them now. If you're not clearly ahead, bide your time.

4. Big or small? Should I target small towns, which will be converted when having 51% followers (right?), or huge cities, where it will spread more rapidly in the in-town population (will it really?)

This is risk vs reward. Smaller cities are easy to convert... but then they are also easier to reconvert. Bigger cities take longer to convert but are harder to reconvert.
It very much depends on how close and active your religious opponents are.

As far as I'm aware, big cities do not accelerate their internal spread to the rest of the city's pops. However, when big cities get converted, they do accelerate the regional spread of religion due to religious pressure.