Something has to be different than what you say for this scenario to occur in a house that's wired in a manner even approaching code compliance.
The voltage drop on the outlet while the fan is on is normal. The fan is simply using some of the power of the circuit, decreasing the remaining potential elsewhere in the circuit.
You say that the voltage changes at the outlet when the fan is on, but the outlet does not work. This means you are getting voltage at the outlet even when the fan is on. Are you reading anything close to 120V? (anything between 110 and 125v is "normal" household voltage in the U.S.). Do you get significantly more voltage when the fan is switched off? If this is true, my hunch is that the branched connection at the J-box served by the panel is not as continuous as it should be.
With the circuit deactivated at the panel, take apart that junction (hot and neutral), make sure the wires all have about 3/4" of insulation stripped, the wires are all twisted together (you can't rely on the wire nut to secure an extra wire to an existing twisted wire junction), and the wire nut is securely screwed on to the twisted bundle. Tape around the entry to the wire nut for both the hot (black) and neutral (white) wires, then before you put everything back, turn the circuit on and try your multimeter again with the fan off and on.
If it's still cutting out, make sure all the connections in the switch and in the ceiling are kosher in much the same way.
Just a shot in the dark; are you sure the switch isn't a three-way? If it is, then the two terminals at the bottom of the switch are not continuous; you can't hook on to either of them with your new hot wire or with the wire from the panel to the J-box you're hooking into. You should instead hook into the top (Common) terminal (it will be the only one on that end of the switch body), which should be fed from the panel and will thus take the switch out of the circuit path to your outlet. If making that change means that something else is now always on, then you need to find or make a different branch of the circuit for your outlet that is fed directly from the panel.
You have a new-style switch loop -- the current (2014) NEC now requires switch loops to have a neutral, not just a hot and a switched. Switch the black and red wires on the light, and the white and red wires on the switch -- this means the light goes to red and white, and the switch goes to black and red, with the white wire in the switch box capped off with a wirenut.
(The reasoning behind the new requirement is to allow future timers, dimmers, and other such "smart" switching devices to be designed in a more sane manner where they can draw power for their own operation in a normal way, instead of trying to turn their load on slightly at all times or sending that working current back through the equipment grounding system.)
Best Answer
The light and the fan are on different wall switches. If these are on different legs (180 deg out of phase), then the net current in a common neutral in the fan will be the difference in the currents in the two hots.
IIRC when the light kit is installed on the fan there is a wire nut splice of the neutral of the light to a common neutral (whites), and of course there is a splice to the hot of the light (blues).
Suppose the fan and light have a common neutral in the fan and there is significant resistance in the neutral wire or in the splices where the light and fan neutrals join the common neutral in the light fixture or where the common neutral in the fan shroud connects to the neutral of the house wiring. When both the light and the fan are on the voltage at the fan motor and the light sockets is higher than when only one is on.
Easiest way to test this explanation is to remove all the bulbs but one from the light fixture. With the light off turn on the fan and let it come to constant speed, then turn on the light. The fan should speed up much less when all the bulbs were in their sockets.
If you can reach both of the wall switches with voltmeter leads, test the voltage across the two.hots. Zero volts indicates that the two hots are on the same leg and this explanation is wrong. 240 V means they are on different legs and this explanation is possibly true.
The remedy would be to redo the splices where the neutral come together. If the neutral wire itself is at fault, then the wire is damaged or you are using higher power bulbs (high power incandescents) than the fixture is rated for.