The led strips can be powered from wither side. Only the data signal has a given direction.
Does the typical application circuit (last page of the datasheet) indicate that all leds are running in parallel? Does this mean I don't have to connect every single strip in series by connection the + and - from the last led and just connection it conveniently to the source or will I run into issues?
You can and should (for long strips) run power in parallel to help with voltage drop. But I've not yet encountered any problems due to voltage drop for upwards of 300 or so LEDs. Use a good power supply.
So really, wire it up however is convenient. You'll do no harm either way. It's just a simple power bus. Worry about it only if you actually have a problem.
I personaly just ran into a lot of trouble doing such a parallel power connection.
I have around 450 LED on 17 stripes, and they are all powered in parallel from the middle of each stripe.
I've seen that first LED of each stripe eventually died (around 10 LED so far). I'm not sure, but I suppose it's because of the difference of potential at the end of the strip between the grounds. If a stripe is lit up, and the next one is not, there could be a difference.
As the ends are only connected through data line, I believe my chinese stripes are very sensitive, and there was a difference of ground.
I've now connected also the grounds together where the data lines are connected. They are still powered (large wire) by the center of each stripe in parallel, but the additional chained ground connection at the extremities (thin wire) seems to help since I've not damaged a single LED since.