How do I ensure my first row of hardwood flooring is straight?
I'd like to write an answer on the overall methodology on how to pick where to align a floor, but I don't have the chops for that. But in your case, I do have a suggestion.

One of the principles of flooring installation is that it's more important to look straight than to actually be straight. For example, you have that architectural detail separating the two rooms. I think it would look more odd if the flooring did not line up with that detail than if you had a wedge shaped piece against a wall. Snap a chalk line down the middle, measure from that line to within a few inches of the starting wall, and snap a second line parallel to the first. Align your first piece with the second line. Then every couple of rows, measure to make sure you are still parallel with the first chalk line.

An alternative would be to snap the first chalk like as I described above and start from the middle of the room(s) and work your way out. Depending on the type of tongue and groove and nailing method, this may not be possible.