The wiring may not be conventional because the power or the light circuits are not coming into the switch boxes and some creative wiring was done. The easiest way to figure this out is to first disassemble all switches and lights to access the wiring. Find out which wires in each box have power and make a diagram. Next shut off the circuit so you can check each wires resistance/continuity. This will allow you to determine which cable is feeding where. With a voltmeter/DVM set to OHMS pick the first wire to check and make sure there is no continuity. With the meter still connected to the wire go to the light or switch box you think the wire is running to and touch the same colored wire to ground (this is assuming it hasn't changed color by a connection somewhere else). The meter will read close to zero ohms when the wire is touched to ground (some wires may read a low resistance if it is connected to the light bulb so you may want to remove them). Do this for each wire until you have a diagram showing where all wires are going/coming from. Then is is a matter of having the wire supplying power go to the common of one switch. The two other wires in that same switch box must get to the other switch. The other switch will have the two wires coming in from the first switch and then the wire going to the lights connect to the common of switch 2. It now is a matter of connecting the wires through all the boxes to achieve this configuration. If you cannot figure this out then a picture of where all the wires go to the switches and lights. Good Luck!

If it were me, I'd go with a remote switch and skip the wiring all together. We have that in our bedroom, and it works fine. You can mount the remote to the wall near where you'd put the switch. That fan came set to be used without a hardwired switch. The wires are in place if we ever changed it. We do have a remote wired in (so it does not need batteries) where the switch would go, and a second one on the headboard. We installed remotes for our two kitchen fans also - so we can turn the fan on and off without using the pulls as that gets cumbersome. These two did not come that way, but it was easy enough to put the controls in the motor housing. We still use the switches to control the lights. (the south is on a three way switch and the north is on a four way switch) The big fan we have in the library came with a remote, but we wired it to a main of/on switch also. This one is kind of a pain becuase the switch controls the power, but the remote controls the dimmer and fan speeds.

NEC 200.7(C) Circuits of 50 Volts or More. The use of insulation that is white or gray or that has three continuous white stripes for other than a grounded conductor for circuits of 50 volts or more shall be permitted only as in (1) through (3). (2) Where a cable assembly contains an insulated conductor for single-pole, 3-way or 4-way switch loops and the conductor with white or gray insulation or a marking of three continuous white stripes is used for the supply to the switch but not as a return conductor from the switch to the switched outlet. In these applications, the conductor with white or gray insulation or with three continuous white stripes SHALL BE PERMANENTLY REIDENTIFIED to indicate its use by painting or other effective means at its terminations and at each location where the conductor is visible and accessible. . http://freenec.com/T21.html

Is there a hallway wall behind that ties into that stud? If so, not a good idea to remove the whole stud, maybe just remove a portion of it? And add blocking to tie it into the next stud to the right? What about moving the switch box farther into the room, as in attaching your 3 gang box to the stud to the right of your arrow? Or use a 2 gang box with dual switches to save space? https://www.homedepot.com/p/Leviton-Decora-15-Amp-Single-Pole-Dual-Switch-White-R62-05634-0WS/100356887