I notice that most weapons other than the original weapons have a level above 1. Can this number be different from player to player (or even item to item in the backpack of one player) for the same item? If so, does it have any effect in game?
Most items have a set number -- before the Scout update introduced the Sandman, no item went higher than level 10.
About the time of the Spy / Sniper update, there was a glitch in the random drop system, and normal weapons (that usually have a fixed level number) were generated with a random level between one and one hundred. This has since been fixed, so if you see, to use one of my own items for example, a level 56 Dead Ringer, that means the item is very old. (And now with the latest patch, you'll never see a non-vintage oddly leveled item)
On the other hand, hats will always have a random level. This means nothing, but certain levels are certainly more valuable for trading, for pure sentimental value. (1, 100, 69, 42, etc.)
The singular exception to all of the above is the new Dueling Badge medal misc item, which actually "levels up" as you kill folks during your duels. A level 100 dueling badge works out to 1000 kills (Might be off a factor of ten) during duels, and likewise the rarity of the badge itself (bronze, silver, gold, platinum) is based on level increment. 0-25,26-50,51-75,76-99,100, etc.
In no case does the level of an item have any effect on gameplay.