Thought experiment about the outskirts of the universe Suppose I aim at some object that is 13 billion light years away from Earth. Let's assume I have a twin that pilots a ship that travels in infinite speed, meaning that he can reach that object  and get back here in a split second. On the other hand I'm the poor brother, and I own a ship with top speed of only at 99.9% of c.

We leave earth at the same time, with the same destination. My ship accelerates at constant rate for a full day until we reach max speed, then we enjoy the travel without accelerating for a second time.

1) how long will it take me to meet my brother, and how older will he be? 
2) what will be the age of the universe, according to me? 
3) what will be my current speed relative to surrounding objects before decelerating? Would I be able to accelerate and reach max speed again? If yes, what's the point of accelerating later rather than sooner, and having a top speed after all?

4) And a plot twist: suppose that while in travel I build a second ship exactly the same as mine. When we're about to arrive we choose a distant point 13 billion light years away in the same direction, and I send a friend on it, who accelerates until he reaches 99.9% of c relative to me. If we repeat that for N times, will we ever reach "free space" where matter didn't reach yet, or some other cosmologically interesting place? How does it look like there? If not, a) can we tinker with any parameters in order to do it "right"? b) any other plans on getting there?

I understand relativity from what I learned in college (long time ago), is there any other theory needed to understand such traveling questions?
I will answer all of your points with zero tact and/or precision in order to fit them all in.

1) It will take you about a "split second" given you stated that it only takes your brother a split second to travel from Earth to the point 13Bly away and back again. He will be exactly said split second older and no more, because as he is traveling faster than the speed of light, the rules governing his and your aging no longer apply, meaning you have created a scenario where both his and your time's are connected and simultaneous. 

2) At the point when you meet up with your brother a split second later, the universe will be a split second older... At the time you've reached your destination the universe will be [ > 13By + (the age of the universe when you left) ]. The fact that your ship takes one year to accelerate to and from the top speed is irrelevant given the duration of the travel time.

3) .999c ... Why not, assuming that you have enough "fuel" your imaginary ship should be more than capable than making a single one way trip. There given the distance and time it would take you to get there, there is close to zero reason to accelerate either sooner or later, although the whole near speed of light travel is sort of the point of your question, it would kind of be necessary. 

4) The N times scenario is redundant and a little weirder than your other questions, as the speed of light is the same in all reference frames, you would never breach that boundary, however to each previous ship the rate of travel would appear exponential even though there would only be infinitesimal gains. Would you ever reach free space... no, without faster than light travel it is not possible. I would venture a guess that space would look somewhat similar to the here and now regardless where you ended up within your lifetime. a) by tinker, you could assume that you were able to travel as fast as your twin in part (1). b) Think Star Trek, impulse power was sub light speed travel, all warp speeds are some large multiple of c. 

Look into relative simultaneity and space-time diagrams for reference on the whole twin paradox issue. Also you could read something like ... the physics of star trek for faster than light travel scenarios with actual physics explanations or problems.