That 15A breaker and the trip
Remember this: Breakers protect wires. Breakers come between you and wires overheating and starting a fire in your walls where it's almost impossible to put out.
Code is very clear: #14 wire gets a 15A breaker. #12 wire gets a 20A breaker. (NEC 240.4). You are not allowed to put common receptacles on a larger breaker (210.21B3). That is because breakers also protect appliances. A 15A breaker allows 1800 watts (20A allows 2400W).
Appliances which make heat (and have standard plugs) are hard capped at 1500 watts (UL) - and that's not very much, so most heat-making appliances use all of it.
People have no concept of what appliances actually draw -- you use the word "little" to describe the heater, as if that makes it a small draw -- it's a 1500 watt draw. Go look :) Your coffeemaker is probably around that too, since it also wants to heat the water as quickly as possible.
If you're noticing, two heat appliances is 3000W and won't even fit on a 20A breaker. Correct. Heat appliances are like Sauron; they do not share power!
If you need to run 2 large loads, run 2 circuits or position them so they are on different circuits. No way around it!
Some fool had the same problem, and that's why he fit the 25A breaker. Why did it work then, since 25A=3000W and your draw was slightly above that? Breakers operate in "thermal trip" mode, where they allow the overload, allow the wire to heat up, and only trip before the heat becomes dangerous. This scheme depends on the wires being the proper size.
It also depends on Do not immediately reset a breaker after a thermal trip. Wait 10 minutes for the wires to cool off, since breakers cool off faster than wires. Well, okay... the first trip, go ahead and reset immediately but remove some large loads.
"Equal currents" is a basic principle
As you know, current flows in loops. (kinda like hydraulic fluid). Out the hot, back the neutral. These partner wires must travel together in the same cable or conduit. If power goes out to a lamp, the return must come back on the same cable. Why? A bunch of reasons.
First, Unlike hydraulic fluid, electricity throws a large magnetic field - and because it's AC power, this field vibrates. If the wires are tightly bound to each other, the fields cancel each other out and the effect is null. But if the wires are separated at all, it causes physical vibration (damaging the wires) and eddy current heating in metals.
But second, neutrals don't have circuit breakers. It is supposed to be protected by the idea that current on a neutral can only come from its partner hot wire, which does have overcurrent protection.
Third, GFCI detection requires this - they check whether currents are equal to see if any current is leaking or shocking someone.
You are allowed to split; say for a ceiling fan, you can split power to lamp and fan, both returning current on the same neutral. The 3 wires add up, and cancel each other out.
You're usually safe if you physically run cables in a tree fashion -- branches can branch and re-branch, but they can't loop back to another branch.
And there will always be at least 1 partner wire. And the pair (3+?) will always be the same size as each other.
So this neutral wire is an anomaly
Since the electricians looked for other cables in the back of the box, we know this must be a weird single wire that's been bridged over from somewhere else. No doubt that circuit had a problem with its neutral wire, so they "stole" neutral from here. Great: if it's a 15A breaker, then this circuit's neutral can carry 15A + the 25A from your breaker = 40A. That's not good!
My thought is, figure out which neutral is the partner to the hot, and remove and cap off the other one. Treat it like it might be hot; it will be once you remove it, because it will get pulled up to 120V by the appliance trying to return current.
If a nearby switch has a powered switch, it's possible that switch is wired as an old-style "switch loop" and he stole neutral from here. Roll it back to a switch that doesn't require neutral. For instance timer switches that crank (instead of pushing buttons) don't need neutral.