Electrical – Could running two 15 amp circuits of NM wire in parallel cause interference

I believe that it would be against the NEC to have these two seperate circuits with a shared neutral and not have them on a double pole breaker (or at the very least a handle tie between a 15A and a 20A breaker).

Branch Circuit, Multiwire. A branch circuit that consists of two or more ungrounded conductors that have a voltage between them, and a grounded conductor that has equal voltage between it and each ungrounded conductor of the circuit and that is connected to the neutral or grounded conductor of the system

210.4 Multiwire Branch Circuits. (B) Disconnecting Means. Each multiwire branch circuit shall be provided with a means that will simultaneously disconnect all ungrounded conductors at the point where the branch circuit originates.

A double pole breaker will simultaneously disconnect both circuits and satisfy the code requirement.

20A breaker is required and 12 gauge copper 10 gauge aluminum is the minimum depending on how long the circuit run is.

Furthermore even if it is on a 20A circuit, you are not required to upgrade your 15A outlets to 20A outlets. This is only required if there exists a single receptacle on the circuit. One can only plug in 15A max devices into a 15A outlet, therefore the 20A load circuit load would not be achieved on a single device.

It might also be a code requirement now for a detached building to have its own grounded sub-panel as well but that I will have to look into.

You are correct that the first two options are allowed by Code (citing from the 2014 NEC here) -- the relevant passage is 210.11(C)(3) along with its Exception:

(3) Bathroom Branch Circuits. In addition to the number of branch circuits required by other parts of this section, at least one 120-volt, 20-ampere branch circuit shall be provided to supply a bathroom receptacle outlet(s). Such circuits shall have no other outlets.

Exception: Where the 20-ampere circuit supplies a single bathroom, outlets for other equipment within the same bathroom shall be permitted to be supplied in accordance with 210.23(A)(1) and (A)(2).

However, this passage only applies to receptacles unless you take advantage of the terms of the exception; putting the bathroom light on a shared lighting circuit is Code-legal as long as the receptacles are on a dedicated circuit.

For the heaters, you can either put them on a per-bathroom circuit as long as they satisfy 210.23(A)(2) by not using up more than 50% of the circuit's ampacity:

(2) Utilization Equipment Fastened in Place. The total rating of utilization equipment fastened in place, other than luminaires, shall not exceed 50 percent of the branch-circuit ampere rating where lighting units, cord-and-plug-connected utilization equipment not fastened in place, or both, are also supplied.

or you can have them on a shared lighting circuit, if you decide to have a separate circuit for bathroom receptacles. Unfortunately, putting them on the same circuit as the bathroom receptacles when that circuit is shared between bathrooms is prohibited by the Exception to 210.23(A):

Exception: The small-appliance branch circuits, laundry branch circuits, and bathroom branch circuits required in a dwelling unit(s) by 210.11(C)(1), (C) (2), and (C)(3) shall supply only the receptacle outlets specified in that section.