There are standard tuning SysEx-es (of course, not all devices understand those).
You can add a SysEx into a MIDI file and it will remain a MIDI file.
.syx files only contain SysEx-es, so if you try to convert .mid to .syx all non-SysEx messages like NoteOn will be lost.
SysEx is by definition specific to a particular system. Each system may be able to do different things, in different ways, using SysEx. There are a few 'global' commands that have become accepted, but you cannot assume that ALL systems will recognise them.
The minor adjustments to tonality that you are referring to could well be capable of being set/adjusted by making changes to specific bytes in a device's memory, and this might usually be done via the 'front panel'' but could be done byt SysEx, assuming that the MIDI Map for the device specifies the memory locations, and you can create a SysEx command that addresses the relevant memory locations and changes data there. In effect, you're doing what the 'front panel' operation would do.
This will be dependant on the specific device, the memory map, and the SysEx implementation of that device.
The same would apply for most other midi commands, but I doubt that it would apply, for example, to basic things like Note On/Off.
If the device has a front panel facility to do x, by pressing a button or turning a wheel, then it may be possible. Other things may be possible ONLY via normal MIDI.
Which specific device are you planning to use. If you said, maybe I might find details of it's MIDI implementation.
The MIDI Tuning standard is a set of System Exclusive messages to set fine tuning of each MIDI note number. You can put MIDI Tuning standard messages into a MIDI file along with notes. It is still a .mid file. Details are in the MIDI 1.0 specification here: https://www.midi.org/specifications/midi1-specifications/m1-v4-2-1-midi-1-0-detailed-specification-96-1-4
Technical Standards Board, MIDI Association