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.
This is compliant...
The 110.26(E)(1)(a) rule about dedicated space goes as follows:
(a) Dedicated Electrical Space. The space equal to the
width and depth of the equipment and extending from the
floor to a height of 1.8 m (6 ft) above the equipment or to
the structural ceiling, whichever is lower, shall be dedicated
to the electrical installation. No piping, ducts, leak protection apparatus, or other equipment foreign to the electrical
installation shall be located in this zone.
Note that it says that the dedicated space is equal to the depth of the equipment. Since the box is surface-mounted to the plywood, this means that the dedicated space starts at the front of the plywood sheet (the side facing out towards you), and the piping behind the sheet does not infringe upon it as a result.
Personally, I'd deploy a NEMA 3 family enclosure here instead of a NEMA 1 to minimize the risk of problems due to splashing, etal, but that's just me being mildly paranoid.
but be careful where you put the washer!
However, the Code compliance of this also has to do with where you put your laundry appliances. The washer needs to be placed in compliance with the 30" minimum width requirement of 110.26(A)(2):
(2) Width of Working Space. The width of the working
space in front of the electrical equipment shall be the width
of the equipment or 762 mm (30 in.), whichever is greater.
In all cases, the work space shall permit at least a 90 degree
opening of equipment doors or hinged panels.
Note that this doesn't have to be centered on the equipment -- in your case, simply making sure that the washer is 30" away from the side wall and not putting anything else there(!) would be enough to comply.
Best Answer
The good news is that your proposed closet is big enough
The good news about your plans is that your proposed electrical closet is indeed large enough to house the clear working space for your electrical panel (30" wide by 36" deep, though not necessarily centered on the panel), as well as a future subpanel adjacent to the existing panel and sharing its working space, as long as the existing panel is 6" deep or less, which is true for just about all residential loadcenters.
Keeping pesky storage out ("Do this, Don't do that, Can't you read the signs?")
The main issue with your closet plans, like with dedicated electrical closets in many commercial spaces, is the penchant for people to not know that the NEC means what they say when they talk about clear working space for panelboards. In fact, NEC 110.26(B) explicitly says that electrical working space shall not be used for storage:
To this end, a sign on the closet door reminding you and yours that it's an electrical closet and not a storage closet:
would be wise; these signs are available both as vinyl decals for use on solid doors, or as aluminum signs that can be screwed to a solid or slatted door, and aren't all that expensive in the grand scheme of things.