Firstly, let me apologise for what happens next as it's a while since I've flexed the muscles of my engineering degree.
To tidy up your first statement, compressing air does not increase it's mass. Also, discussing pressure change across jet engines can get confusing rather rapidly so I'll try my best until someone better steps in.
IIRC, total pressure is the key, and in particular, keeping as much as possible through compression, expansion, etc.
1. Air inlets are designed to present non-turbulent air at a manageable speed to the compressor. Ideally, no work is done and thus total pressure is unaffected.
2. Both axial and radial compressors are doing work on the air and thus the total pressure of the system will increase. If 100% efficiency is assumed, this air is passed across a turbine and imparts all of it's energy as work, thus leaving the system in equilibrium, hence no thrust. Combustion at high pressure is used to fix this.
All is neatly covered by a temperature-entropy (T-S) plot, which sounds horrid but is the one thing that underpins most of this and does not require degree maths to interpret.
Ultimately, if total pressure is constant, a change in dynamic pressure will drive a change in static pressure. If it is not constant, i.e. work done on/by the air, both may change.
So we have:
1. I should read my books more often
2. Mass does not increase, but total pressure changes as it does work, and is worked on.
3. TS plots are useful, try one at this link until a more meaningful answer is posted.
4. You WILL get a better answer if you put this on the Tech forum.