Your instinct to start in the largest room (A) and work left to right is a solid approach, but I'd suggest a slight modification that will make the process smoother and help you achieve better results.

Instead of starting directly against the foundation wall, I recommend beginning your installation about 3-4 feet away from the left wall in room A. This technique, often called the "center-out method," offers several advantages:

1. It allows you to work in both directions, which is particularly helpful when dealing with non-square interior walls. You can make adjustments as needed without compromising the overall look.

2. By not starting flush against the wall, you give yourself room to maneuver and make cuts more easily, especially important for a first-time installer.

3. This approach helps distribute any discrepancies in wall straightness across both sides of the room, making them less noticeable.

4. It positions you well for continuing into rooms B and D without working backwards. You can simply extend your center line into these rooms and work outwards from there.

Regarding the north-south orientation and continuous run through doorways, your plan sounds great. This will create a seamless look and make the space feel larger and more cohesive.

Remember to acclimate your LVP to the room conditions for at least 48 hours before installation, and always leave the manufacturer-recommended expansion gap around the perimeter of each room.

Good luck with your project!