WRONG - At low level EAS equals TAS
CORRECT - At ISA Sea Level EAS equals TAS (and CAS).
WRONG - At height EAS is lower by the density factor
CORRECT - At height EAS is lower because of reduced Compressibility which is directly related to Mach Number
WRONG - Thrust is also lower (at higher altitudes)
Depends upon the altitude. As you are at a constant TAS, CAS/EAS/Mach are reducing, so far so good, until you climb higher and the EAS/Mach is below Vmd/Mmd, then the required Thrust will increase.
Does it work out that the same N1 gives the same TAS at all heights or would you have to adjust N1 up or down?
Depends upon the bypass ratio of your engine, i.e. the ratio of thrust produced by N1, and the thrust produced by the core (N2/N3). Too many variables here.
Regards,
Old Smokey