OK, first of all, plainly this is a mess. The fact that there is a white wire hooked up to a black wire without any recolouring of it alone says that (1) there's some amateurish stuff going on here, and (2) you cannot trust any of the white wires to actually be neutral until you prove that they are. So proceed cautiously, and take notes as you go.
There are a lot of questions here but a good one is:
What's the easiest way to track a specific wire to see how it flows without ripping out the walls?
Start by obtaining a long three-prong extension cord. Plug it into an outlet that is on, and verify with a voltmeter that there is voltage between the hot and neutral, there is voltage between the hot and ground, and no voltage between the neutral and ground. You now have an extension cord which you can use to provide a known ground and a known hot.
Next, turn off the power to everything that you're going to be working on.
Next, put a piece of tape with a number on every wire that you're going to be working on, and start making notes about the color and location of each numbered wire end, and what they're hooked up to now. Take photos. You want to be able to put this back together the way you found it later. Number the switches as well.
Next, carefully get all the wire ends exposed but not touching anything, and turn the power back on. Use your known ground and your voltmeter to determine which wires are hot under every combination of switches. Remember, some of the whites are hot.
Turn the power back off again.
Test everything again to make sure that no, really, the power is off.
Now test everything again against the known hot to find out which wires are neutrals that run back to the panel. Again, test everything against all combinations of switches, remembering that at least one mistake has been made in this wiring so far; there might be more. Interrupting a neutral with a switch is almost always wrong, but I have found plenty of interrupted neutrals in old houses.
Now unplug your extension cord from the wall and get a flashlight. Rig yourself up a continuity tester out of the extension cord and the flashlight. (Of course if your voltmeter has a continuity tester, use it instead of jury-rigging one out of a flashlight.)
Suppose you suspect that wire ends labeled 2 and 9 are the same wire. Attach end 2 to the extension cord ground. Run the extension cord to the other. Now wire the extension cord to the battery, the battery to the light, and the light to end 9. Did the light go on? Then your hypothesis is confirmed. Did it not go on? Try every combination of switches. Again, record your observations.
Keep doing that until you have enough observations to form a consistent theory of how the wiring runs through the walls. Draw a diagram.
Once you're there, you should be able to solve your problem. If you cannot, then you'll be able to post a question here with a lot more information.
Also, take this opportunity to ensure that every wire is correctly coloured, by putting tape on miscoloured or confusing wires. Also, if there are oddities such as a box that has hot wires from two different breakers, make a note of that for the next guy.
Looking at your diagram, you are using the white wire as hot to the left-hand switch and black and red as travelers between the two 3-way switches. On the left-hand switch, you will find (excluding the green ground screw) one terminal looks different, probably a different color) than the other two. This is where the white wire goes. The black and red wires go to the other two terminals. Your diagram implies that this is wired wrong as you show the black wire going to the separate side of the switch.
Two more comments:
(1) I’m assuming that you also have ground wires running everywhere that you just haven’t shown. If not, stop and get a professional to help before you kill someone!
(2) You should add black tape to the ends of that white wire to show that it is not a neutral wire.
Best Answer
In the most basic way it would depend how the wires were fed between the lights and the boxes. There are many combinations so we would need to see a diagram of were the wires go.
So you would need to do a drawing and post it. If you do that somebody would be happy to draw ontop of it for you and connect your wires if it can be done. (Do not worry about drawing the grounds, we will assume all your boxes have them). To test where they all go you would need to
TURN off your circuit and test that it's off.
Than you would need to open all your boxes and open all the wire nuts except the grounds. Leave them connected. And make sure no wires are touching each other. Now you need to set your meter to the low ohms (omega symbol) OR continuity setting (looks like 3 lines ontop of each other projecting sound). Now you may go around from box to box piecing together where the wires go. Go into the first box and attach a red or black wire to a ground (have them touching) and with your meter in the second box put one lead of your meter on any black or red wire and one lead on a ground (all the grounds are connected so any one is fine).
All multimeters are different but some multi meters have a continuity function , where it beeps if there is continuity. Basically it beeps if those are the correct wires and current can travel that path around the meter. Meaning that is the same wire/ same cable on whatever two wires your meter is on. If there is no continuity it will not beep. Other meters to show there is continuity will have a very low ohms reading. If it wasn't a match it would have an infinity number or OL.
Once a wire from each cable is tested, record those findings and make a drawing. You will not have to say where each wire connects on the drawing (we more care where each cable runs to and how many wires). But maybe you want to draw that so you can put it back together!
We will also need to know which cable is the home run/ hot power cable. To test that with all the cables still NOT touching (except grounds with each other) you will need to flip the circuit breaker on. And very carefully with your meter on Volts AC go between what you think is a hot (either black or red) and a ground and see if you have 120V. If you do. Thats the home run and record it on the drawing. That is an important step to let us know that. Don't forget to write which boxes are switches and outside lights etc... Good luck.