to which device is matrix damage applied?
Book doesn't really say, but if your matrix persona is targeted then I would rule that Matrix damage will be soaked by the device you are using to access the matrix with.
For regular users, that normally use a commlink (+ datajack or trodes), this will be the commlink.
For riggers, that normally use a RCC (+ control rig), this will be the RCC.
For deckers, that normally use a cyberdeck (+ cyberjack), this will be the cyberdeck.
For technomancers, that normally use a living persona rather than a device, matrix damage will instead be soaked by their regular condition monitor(s)
do I really have less program slots with a cyberjack than with a commlink?
Book doesn't really say, I would rule that program slots are provided by the device you are using to access the matrix with.
For regular users, that normally use a commlink (+ datajack or trodes), this will be the commlink.
For riggers, that normally use a RCC (+ control rig), this will be the RCC.
For deckers, that normally use a cyberdeck (+ cyberjack), this will be the cyberdeck.
Technomancers, that normally use a living persona, don't have program slots (but they might learn echos to emulate specific programs)
Is it possible to "split up" the movement of one movement action
Book say that you are not allowed to take more than one movement related action (Avoid Incoming, Move, Sprint) per round. You are for example not allowed to move up to 10 meters, take a shot and then move up to 10 meters again (or avoid an incoming AoE later in the round after you already used a Move action).
But it sound more like you are asking if you are allowed to move [out of cover] to take a shot and then move [into cover] again while not using up more than a total of 10 meters. While this might sound harmless I would
only allow this If and Only If it is not being abused.
Let me expand on that a bit.
IF you choose to allow this THEN you need to be aware of that your players can start to run out of cover to fire their gun and then run back behind cover before opponents get a chance to fire back. And then GM will have the NPCs start doing the same. Then you will get very strange fire fights where nobody actually have a direct line of sight to anyone ever

By RAW this would be either be resolved by moving [out of cover] with the Move minor action (or perhaps Stand Up minor action depending on situation), taking your Attack major action and then you finish your turn with a Take Cover minor action OR by staying in cover while taking your Attack major action (by either spending a minor action or two edge to Fire out of Cover and not earning edge on the attack).
Not by taking a Move minor action "that you split in two".