If a player is absent and nobody chooses to play this absent character and instead I let it vanish for a while with a suitable reason or excuse, does this missing player gain levels and XP with the other players or does his level only increase when he plays again?
As none of the previous questions ask for 4e specific advice, let me give you some.
Overall just use common sense when dealing with absent players. The most important thing is to not let the players who can make it get ahead of the players who cannot, it's not really fair for those of us who have life intervene and it will likely cost you players.
You've got to give the absent character XP. Players whose characters have fallen behind are less likely to show up at the next session.
Remember, XP is not a reward for playing the game. Playing the game is its own reward.
It's best to write the character out. Send him on an errand, have him get sick or arrested. Try to capture the player's interest with the character's situation so he shows up next time: "Korgath got a tip on that Holy Falchion he's been looking for and risked going into the quarantine zone to get it." or "While praying for spells, Orland was taken up to Elysium to stand trial for cutting that guy's throat last week."
The answer to your question depends on several things: your game style; how badly you want to accommodate your players; and ultimately your personal taste as a DM.
In a game where there is a high risk of death or that is very dangerous, it is probable that a character two levels below everyone else will die, or the other players will not be challenged by the encounters. Also, if a game is difficult in general, a weaker player may not be able to do much in sessions. This means that denying a character xp lowers his usefulness and survivability. I'd suggest simply giving the absent character enough xp to keep up with his fellows, and keeping any penalty small.
However, if your game is more 'friendly', a lower-level character might be less powerful than the other players but still able to have an impact on gameplay. Over time, a lower-level character will catch up to the others. Losing out on some xp is much less dangerous. In this case, you'll have to decide based on the other factors, although simply leaving the character as-is while the player is away is much less of a problem, especially if they have an ability that means they'll still be useful if they are a lower level (e.g. some skills, all healing, most buffs, a lot of magic/ranged attacks).
This is fairly self-explanatory. If your player wants above all else to be the most powerful, by stopping his character from leveling up, you will either be driving him away, or be forcing him back into the game to keep up, so use this only if you want to change his behaviour. A player who only wants to be 'one of the group' would probably prefer to stay at about the same level as his companions.
It is rare to encounter a player who objects to xp, but it is possible. If you plan on leveling up their character, always talk to the player first just to let them know. Some roleplayers, for whom the character is more important than their abilities, might want to be there for everything that happens to him.
I can't give too much advice on this, because I don't know what your player(s) are like, but I'd suggest talking to them, as it involves their character, even if it's only to give them a heads-up of what's happening. Also, treat all players equally, or have a system of rules, so no-one feels mistreated or as though you have favourites.
At the end of the day, the game world is yours and you can do what you like. Any of the problems mentioned in the 'Game Style' section can be stopped by you playing with how your world works. If you have made a decision, that that's your decision, and the players will have to like it or lump it. However, it's a good idea to remain diplomatic towards players. Once you've made your decision, you should explain to the player why you've done so. If they complain, allow them to present their case, and then explain why you will stay by your first decision if you aren't persuaded by them. It's your game, after all :)
There is really no set answer for this question, and every group has to decide for themselves. The most common methods I've seen were these.
The character is absent entirely. In more episodic campaigns, the character can leave for some business of his own, or be busy with something related to the adventure, but not interact with others directly (the party thief going to scout ahead while the others travel with the supplies, for example). If it's interesting enough and you have an opportunity, you can have a solo session with the player, or use the character's absence as a hook for future adventures.
Another player or the DM plays the character. Only do this when the absent player agrees. As a bonus, this lets you use the character to give other players hints if they get stuck.
The character is there, but fades to the background. Technically controled by the DM, but doesn't do anything of importance (unless his unique skills are required or he could start a conflict within the party, although you'd probably want to avoid such situations) and generally just follows the party around. Not always appropriate, for example in combat-heavy sessions.
Cancel the session. Certainly not pleasant, but if the character is important for whatever's going to happen during the session. Run a one-off or dust off that Twilight Imperium box you have lying around. If the campaign permits, you can play a one-session sidestory for the rest of the characters.
As for the more mechanical side of the issue, see wax eagle's answer.