There is time-space thing. Moving through space at the speed of light and you don't move through time. Stay stationary and you don't move through time.
Neither of these are true. Moving through space at the speed of light means that you are massless, and will always move through space at the speed of light, in any reference frame. Staying stationary in a particular reference frame means that you don't move through space in that reference frame, though you may move according to other reference frames. In neither case do you stop "moving through time". Time passes while an object is stationary, and time dilation applies to objects that are moving in your reference frame, which may or may not include you. There is no valid reference frame for an object moving at the speed of light.
Q1: Does that imply that there is a point in space, that can be considered stationary for our visible universe, where the clock goes faster than anywhere else?
No. In fact, one of the most fundamental postulates of relativity is that there is no reference frame that you can unambiguously declare as "stationary." If we suppose that there is some absolute stationary reference frame that we should compare all others to, the problem is that there is no experiment you can possibly do that can distinguish between moving at a constant speed relative to this frame and being stationary in that frame.
(I understand that the universe is expanding in every direction, but cannot relate it to Q1)
Well, it doesn't relate to Q1, so this is not a problem.
Q2: Our galaxy is moving through space. Does that mean that we have some baseline value for time dilation in relation to the point from Q1 (if it is valid)
No, because you have to measure the motion of our galaxy relative to some other thing, and which particular other thing you choose doesn't have any fundamental significance. It could be the center of the Virgo supercluster, or the reference frame where there is no Doppler effect on the cosmic microwave background, or one of many other possible references. Each one gives you a different answer for the speed of the Milky Way, and each one tells you a different reference frame is stationary. This is fine, because, as already mentioned, every reference frame is non-stationary relative to some other reference frame. There is no reference frame that every observer can agree is stationary.
Earth is moving through space in some direction with speed VEarth. At the particular time, we have two GPS satellites in opposite locations on the orbit with speed VGPS and on the orbit plane, which is parallel to the direction of Earth movement.
At that time: Speed of the first satellite is VGPS1 = VEarth + VGPS Speed of the second satellite is VGPS2 = VEarth - VGPS.
QMain: If VGPS1 > VGPS2, does that mean that clock of GPS1 (faster speed through space) goes slower than the clock on GPS2 (slower speed through space)
It depends on how you're moving relative to the Earth when you observe this.
If you're standing still relative to the barycenter of the Solar System, the speed of the Earth is nonzero, so the satellite GPS1 is observed to be moving faster than the satellite GPS2 at some instant. This means that time would be observed to be passing slower for events on GPS1 than events on GPS2, both of which would be slower than events for a stationary object in this frame.
If you're standing still relative to the center of mass of the Earth, then both satellites are observed to be moving at the same speed, so there is no difference in the rate of passage of time for them. Both satellites' passage of time is slowed by the same amount relative to the passage of time of a stationary object in this frame.
If you're standing still on top of satellite GPS1, then satellite GPS1 is stationary in your frame, and satellite GPS2 is moving, so you observe that time is passing slower on GPS2 than on GPS1.
You can get any answer you want to this question just by placing yourself in a particular reference frame.
