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.
Do not penalize XP for missing players. There are two reason for this.
penalizing XP discourages the players from returning. If I can't be level with the other players because I couldn't make it for a legitimate reason there is no reason to continue playing and
4e is pretty tightly designed for players to be within a level or so of each other and truly works best when everyone is at the same level. Things like to-hit modifiers, defenses and skill checks scale by half-level for PCs and if there is a wide discrepancy it can quickly become un-fun for some of the players.
Keep them up to date on items. Whether that means giving them items when they are there, or awarding them out of session item awards, items are an important part of maintaining competitiveness with monsters in 4e (having a +1 weapon when you should have a +2 is a strong disadvantage, having a +1 neckslot item when you should have a +3 means you die that much quicker).
Have someone recap the story for missing players. Whether this is another player who keeps good notes, or you as the DM that is completely up to you, but you should fill in players about what happened to the party.
Try to avoid scenes/adventures that feature the PC when they are absent. I know this could destroy your carefully laid plans, but if you've got a plan in place that features the absent player's PC try to delay it until they can make it. Most of the time they will love coming back to the group for an adventure that features them prominently.
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 :)