Can you explain the basic rules of kissing hello and goodbye in France?

Ah, “le bisou” fascinating and scary French habit for most foreigners…
First of all, let’s start with the term; “un bisou” means “a kiss”, but it’s not the proper term, the real word being “la bise,” “bisou” is used mostly by children and couples.

So when do we kiss hello and goodbye and we don’t we?
Well, there are no rules set in stone, and he’ll depend mostly on closeness to the person, age, and age difference, situations, gender, as well as regions (for the number) and maybe also social class to a certain extent.

Let’s start with the easiest, that is the number of kisses one gives and what is an actual “bise.”
To kiss somebody hello or goodbye you don’t give an actual kiss, it’s more about putting your cheek against the other person’s cheek and really kissing the air while touching the cheek.
How many air kisses and cheek touching should you give?
It really depends on where you are. The norm in most of France is two, and in doubt it’s always better to give two, to not be the one that gets embarrassed when they put their cheek forward for the third kiss that doesn’t come.
Then, some areas of France give three kisses (the only one I know of is south of the Massif Central area, but I’m sure there are more… in anybody want to give more info on that, feel free). Four kisses are also not that rare, especially in the East of France. Personally, I think it’s too much work to just say hello or goodbye.
In Paris, as one can find people from all over France, you’ll also find people that give two, three or four kisses, even though the majority seems to give two.
Once in a while, before they kiss hello, you’ll hear people saying “so, for me it’s two/three/four (select what applies)”, which I always found amusing.

Now, who do you kiss?
The no-brainer is family.
Anybody kisses anybody else in the family, with the exception of two males from extended family that will sometimes shake hands depending on how close they are.
Then, pretty much all friends kiss, as long as one of both people meeting is a female.

Yes, gender matters when you kiss.
Usually, when a kiss should be given, the gender of the two kissers will decide whether the kiss is actually performed or not.
Two females will kiss, a male and a female will kiss, and two males won’t kiss.
But you’ll see two male friends kiss once in a while.
What’s up with that?
They must be gay, right?
Well, not necessarily. Many straight male friends will kiss too. It seems to be that it’s a fairly recent habit (I’d say the mid to late 90’s), mostly common in the South and in Paris.
The rationale behind this is that the two men consider each other as such good friends that they’re family, so they’ll kiss.
I personally always found the habit quite artificial and will do it only as to not offend my friend.

Sometimes co-workers will kiss too, but that’s only if they consider each other as friends.

When do you not kiss?
Well, you don’t kiss people you don’t know. You don’t kiss people that are your superior unless you’re really good friends to them.
You don’t kiss people you’re meeting for the first time; you should shake hands, but only among adults. Teenagers and young adults will usually kiss when they are introduced to each other.

I think that rounds it up, except for the specific questions you asked and the case of children.

So, who is going to decide to change from handshaking to kissing?
Well, that change rarely occurs actually, the only exception is people that meet as adults and become friends after a while, and as you may know if you know France a bit, that situation doesn’t happen that often. But when it does, nobody decides, one of them will suggest it one day, when they’re sure that both will be ok with it.

How do French men in general feel if a woman they barely know goes for the kiss while they’re holding out their hand?
That very rarely happens too. If somebody is holding out their hand the polite thing to do is to shake it, anything else can be seen as rude. What is not rude though, if the situation feels appropriate is to grab the hand, but still go for the kiss, even suggesting it. And if the situation is appropriate, the man will gladly accept most of the time.

If one just crosses the path of a person without engaging conversation, one usually does not kiss, a nod or a waving of the hands will do the trick while saying hello, it’s even a polite sign that indicates that you won’t have a conservation with this person (or a less polite one if you should have a conversation with this person and you are actively avoiding them).

Concerning children, it’s complicated. Once again, it depends on your relationship with the child, as usual, the closer to them, the more likely you’ll kiss (as a matter of fact, it’s really rare to shake hands with a kid, one kisses or does nothing usually) but, especially if one of the parents is with the child, and the child is quite young, you want to leave the initiative to the child, or to the parent to tell the child, as it’s part of his/her learning process with social conventions and all. I’m talking young children here, for the ones that already master the kissing convention, whether you or them take the initiative is of no real importance.

And if a kid gives you only one kiss, it’s because, the kid is usually pretty young, still learning about the whole kissing thing, doesn’t find it particularly exciting (touching people you barely know with your face… ewww…) so will do the minimum not to be reprimanded by their parents. Growing older, and understand the whole thing better, they’ll starting kissing the number of kisses that is usually being given in the area/environment