Can it has a deeper nose-down tendency in some corners of the flight envelope as well? They are installed much further forward than usual.
From fundamental aerodynamics, the answer is definitely yes.
Probably not a concern of note.
I have no inside information re the specific flight test activities for this aircraft but the usual reason for pitch stability problems with engines pushed out in front at low speed with high thrust and high alpha relate to the normal lift force up front of the engine. In this case, the force is at the nacelle lip as the flow turns into the engine. The force is directed up so we see a nose up pitching moment which is destabilising. Same sort of thing occurs with a number of piston to turboprop conversions where the new engine installation sees the prop pushed well forward for basic CG reasons. Again, the flow changes direction going through the prop disc and there is a significant destabilising nose up pitching moment, especially during the missed approach situation.
On the piston switch to turboprop, the usual fix is to incorporate a SAS facility - typically a variable downspring in the circuit. As far as I can deduce from what one can read in the public domain, the MAX original fix was a cheap way to gain a similar effect utilising stuff which was already in the aircraft's configuration.
Unfortunately the game plan was brought unstuck by later mods in the program as indicated in much earlier posts in this, and other, threads on the subject.
I see PEI_3721 is following the present discussion. His background is quite pertinent so we may get further commentary from him or other specialist folks in due course.