You can't have one switch operate light OR fan. Many jurisdictions absolutely require that a switch on the wall near a room entrance, control a lamp. That is to benefit house guests and mainly from the government's perspective, first responders. There's an exception to allow a switched outlet instead, presuming a person is going to plug a floor lamp into that. There is no exception to allow a fan/light locally switched at the fan. (such things can exist, they can't be the only light). That said:
They now make gadgets to solve the "I have 14/2 and wish I had 14/3" problem. Each has a smart switch (typically 1-gang) with 2 controls, and a control module which sits under the fan shroud. The switch communicates with the module either via powerline networking or via wireless, and some of those are compatible with smartphones or integrated home-automation systems as are now emerging in the market.
Is this normal?
No, but it is not especially unusual.
Do I have a problem with my wiring setup?
Insufficient information to tell.
Do LEDs behave this way?
It is a common problem with some types of LED "bulb" in certain situations.
LED replacements for incandescent lights are relatively new so a lot of house wiring and accessories that people have already installed are designed to work with incandescent bulbs, not with LED replacement bulbs.
Some older accessories that need to draw a small amount of power at, for example, a switch location that has no neutral (only live and switched live) do so by drawing a tiny current through the light bulbs controlled by the switch, the current is too small to make an incandescent light bulb glow but is enough to make most kinds of LED bulb glow.
The working part of Incandescent bulbs, (the coiled tungsten filament) operate at 120 V AC or 230V AC and draw around 500 mA current when working.
Each of several working parts of a LED bulb (each individual light emitting diode itself) operates at about 2V DC and draws maybe 8 mA from the 120 V side of it's driver circuit.
So it can only take a very small current to make a LED begin to light up.
Since LEDs are low-voltage constant-current DC devices, there has to be some driver circuitry somewhere to make them work with 120V AC. This circuitry is either built into the "bulb" or is a separate box. There are many different designs for the driver circuitry and some of these are more susceptible to producing the sort of effects you report.
I don't know if induced voltages from long runs of wiring next to each other are enough to start to drive a LED, but there are probably many circumstances where this happens unexpectedly - it isn't a certain indication of a wiring fault. Far from it.
I tested with a voltage sensor, and it does indeed beep for a split second when hold it up to the bedroom light wire while turning on the bathroom light.
This does suggest some sort of inductive transient voltage being generated. Are there any fluorescent lights in the bathroom?
To properly diagnose this issue you will probably need more equipment - or hire an electrician.
Best Answer
You don't have enough wires between switch and lamp to use hardwired controls.
However, you do have "always-hot" and neutral going to each relevant location.
It's time to go with "smart devices". You need a smart lamp fixture (or a plain lamp with a smart module), and a smart switch capable of remote-controlling said lamp or module without the use of hard wiring. Such things either communicate wirelessly, or they induce signals onto the power wires.
It's difficult to give a product recommendation, as these things are blossoming onto the market very rapidly, and the options are ever-changing. (for that reason, SE's format is to focus on durable advice and not to give specific product recommendations as they are quickly outdated.)
I'm quite sure what your 'maintenance guy' did was intentionally misconfigure your wiring so the light misused the safety ground wire as a neutral/return. That is an illegal and dangerous thing to do, and we won't recommend it here. Not least, any wiring problem can put 120V on the light switch cover screws and electrocute you!