I have an existing ceiling fan in a hallway on a 3 way switch. When I turn on the switch both the fan and the lights will come on. Can the fan be rewired so the fan works off the pull chain and the lights work off the switch without having to run a new "hot" wire?
The only way to know for sure, is to pull the fixture down and have a look at the wiring. If power comes from the panel to the fixture first, then this is trivial. If power goes to one of the switches first, then you're likely out of luck.
If you are lucky, and the power does go to the fixture first. Simply connect the black wire from the fan, directly to the ungrounded (hot) supply conductor. Leave the blue wire from the fan connected how it is.
As BrownRedHawk mentioned in a comment, installing a remote might suit your needs. Though you may want a remote at each switch location, which complicates the installation a bit.
Yes, you can but you need to run an extra wire if you don't already have one.
You basically want to connect an uninterrupted hot to the fan motor. Generally your fan will have 3+ wires + ground. 1 neutral, 1 ground, 1 fan hot, 1 light hot. Your fan hot will have to go to a hot (either interrupted with a switch or directly wired, but then you'll need to turn off the breaker if you ever want to turn the thing off (eg. if your pull chain switch breaks)).
Another alternative, if you don't want to run an extra wire and extra switch(es) is go with a remote control like the Insteon system.
Note that the wall switch is switching the whole fixture, so if you change the wiring as you suggest you won't be able to run the fan without turning on the light rather than vice versa.
Most modern ceiling fans already have two pull-chains, one for light and one for fan speed (hi, med, lo, off). I take it yours doesn't. Might be easier to replace than modify; if it's that old a new one might save power too.
Modern ceiling fans are also set up with independent power wires for fan and light so they can be controlled by a remote or by two wall switches. If yours ismisxing the second pull chain I suspect you might have to modify it for that too... or replace it.
Its really simple if your power is run to the fixture, connect the hot wire to the black fan wire along with one of your switch wires. Now take the blue light wire and connect it to the other switch wire. Connect the 2 neutral (white) wires together and then connect the grounds together. The Fan will now always have power and will only operate using the chain and your light will work using your wall switch, meaning if you have your fan running you can turn your light off (wall switch) and it will not effect your fan. These directions are actually laid out for you in the installation instructions on most new ceiling fans. I hope this helps anyone who was wondering.