Can a huge coiled LAN cable have some trouble for transmitting a signal?

I have done extensive testing at work with various cable and the answer is... possibly, but unlikely.

It really depends how tight the coil is - with Cat 6 - at the bend radius that it comes sold as in a box - no problems... but if you do any sort of tight bend, the chances are high that you could cause problems, but, with cat5e, you could tie it in knots and I doubt there would be any difference.

... Tested Cat 5e, 100 meters, 85 ish meters wrapped in a box, 15 meters up to this punishment:

and I saw no difference when compared to a shorter point to point cable.

Theoretically, yes. However I have never seen excessive coiled wire cause a problem. I dont think ethernet really puts out enough power to create a strong enough magnetic field to cause issues. Unshielded power cables have been known to cause problems.

Rolling up excess wire in a coil can greatly impact its electrical properties and cause interference by literally turning them into inductors and antennas. but you will not effect any significant loss in signal quality by putting away an ethernet cable like this., unless you are subjecting it to enough abuse for mechanical stress to become an issue. A common rule of thumb is to never bend it with a radius under four times the cable's diameter. The coil in your image does not look like it is anywhere near violating that.

When a single wire is rolled up, magnetic fields from whatever source induce currents, adding noise to the signal. This phenomenon also occurs in ethernet cables, but the latter has multiple wires inside, carrying currents in opposite directions. Pairs of wires are generally twisted so that 'on average', both are occupying the same physical space and are exposed to the same magnetic effects. The resulting forces cancel, very similar to someone pulling equally hard on both ends of a rope on a pulley.

Rolling up LAN cables like this is perfectly safe.

While in theory there can be a (small) problem, in practice I've never seen it, and others haven't reported a problem either. The fact that the cable is balanced twisted pair eliminates most inductive effects in the coil.

There is a limit, though, to the length of an individual run from hub to device -- 100 meters. Over that and timings get screwed up, even if the electrical signal is solid. So if someone were to attempt this with a much longer spool of wire there could be a problem.