I've taken half an hour just looking through the auction house, and I have taken my own experience and various forum posts into account, and these are my findings:
The maximum number of sockets per equipment type varies. It seems to be independent of item rarity (magic and rare items, legendaries are something special). Instead, it seems to depend entirely on the body type of the item.
These are the maximums I was able to identify:
First of all: Shoulders, belts, gloves, and boots can't have
sockets at all (more on that later).
Helmets, rings, amulets, weapons, and offhands:
Pants
Chest Armor
Legendary items are a special case: They always have a certain fixed amount of sockets per named item, but just for completeness: The maximum number of sockets on legendaries I could find was two.
There's something I have to add about Shoulders, belts, gloves, and boots:
Even though there seem to be no belts with sockets, it is theoretically still possible that belts can have sockets, and simply nobody has found one yet. I come to this conclusion from a simple fact: Shoulders, gloves, and boots have no search option for "has sockets" in the auction house, indicating that it is indeed not possible for them to have one. Belts, however, do have this search option. It is entirely possible that Blizzard just left it in by accident, though.
Another interesting fact about items and sockets:
Items have a fixed amount of random properties (this can differ, even for same rarity and body placement). When an item is dropped, or identified, the contents of these properties are randomly rolled out. Sockets, no matter how many there are in the item slot, take up exactly one of these random properties.
This make for an interesting conclusion: If you're unlucky, sockets may take away important stats from your item. In general, pure item stats will always be higher than you could at to it with gems. For example, perfect strength gems provide +22 strength. In comparison, items of roughly the same level seem to hit an upper cap of around +60 strength. This means that if you, for example, identify a rare chest armor piece and it finds +50 vitality, +20 experience per kill and +1.2% chance to stun on hit, plus one socket, then that last socket could have been a slot for +50 strength, whereas now you can only try to compensate for it with gems.
On the contrary, however, this becomes a nice bonus when you actually want to stack a certain stats. Items can't roll the same bonus stat twice. If we go with the example from above, you now can circumvent this fact by adding another +22 vitality gem into the socket, thus giving you even more life on the item (good for tanks, for example).
This thread discusses all of this (notice Bashiok's post later in the thread, acknowledging these facts).
Best Answer
You are correct that level 60 is the current maximum level. However, as of patch 1.04, any experience earned after reaching level 60 contributes to your Paragon Levels.
Paragon Levels (Source)
Nephalem Valor
Another gameplay change that occurs at level 60 is Nephalem Valor. This doesn't permanently improve your character, but it does allow them to gain some bonuses to gold and magic find that last until they log out or change their abilities. See the link for a full description, but here's a summary:
Edit: As Holger pointed out, switching acts also causes you to lose the Nephalem Valor bonus.
Also, it isn't mentioned ion the site that I linked to, but there is a limit of five times that NV can stack. Killing more elites/champions will continue to refresh the bonus, but it can't stack more than five times.
The linked site also sounds uncertain as to whether changing equipment will reset the buff. I can confirm that changing equipment and/or potions does not reset the bonus.
Another benefit of Nephalem Valour is that each stack of it increases the amount of XP you gain from killing enemies by 15%. This improves the rate that characters gain Paragon Levels.
UPDATE
Paragon 2.0
Paragon 2.0 will still work as an end game feature, providing players with a way to continue earning levels and experience even once their characters have reached max level. There are several major changes to the system:
All experience earned by max level characters will count towards an account-wide Paragon level. This level, rather than the individual character levels, will award paragon points that can be spent on all characters; not just the specific character who earned the experience.
The Paragon levels earned and points awarded to Hardcore and Softcore characters are tracked separately. Since Paragon levels are for the account, Hardcore players will not lose Paragon levels or points even if/when their individual characters die.
Paragon points allow for a great deal of character customization, with specific bonuses that can be assigned to things like attributes, offensive stats including CC, CD, and AS, defensive stats like blocking %, and general properties like Faster Movement and Pickup Radius.
There is no maximum paragon level (though the individual bonuses may have varying hard caps).
The Paragon experience earned by all existing characters will be added into the system when it goes live.
Characters at Paragon 100 do not earn experience currently, so any play time with them prior to Paragon 2.0 is "wasted" in terms of exp gain.
Exp earned by all Hardcore characters on an account, dead or alive, will probably be added into the Paragon level when Paragon 2.0 goes live.
Many questions remain, and the system is still under development, but the basic functions have been confirmed by Blizzard as essentially set in stone.