What tools and techniques are required for moving & re-sloping a cement slab walkway?
The best pry bar model that I know of for lifting concrete slabs is "big" (as in, the bigger, the better). Use a block of wood or something else that's large, to distribute the weight on the ground, for your pivot. You'll likely have to dig out one, if not two sides to pry and maneuver these blocks. Avoid putting too much pressure on a single point, since that can cause the paver to crack.

For the level, away from the house is good if this is draining against the foundation. You don't need much, but keep it consistent by using a level and measuring the rise over the run.

When resetting the paver, don't just build up one edge. That would leave an air gap in the middle that can result in cracking. Don't disturb the ground below the paver other than to remove any high spots, you don't want it to settle after you're done. Just add stone dust to make a level surface. Then replace the paver, wiggle it a little to work the stone dust into a tight contact, and replace any ground that you dug out from the sides.

For weeds, I'd ask over on the Gardening Stack Exchange, but they make lots of weed killers. You can also place a barrier in the ground, or just go with hand pulling. I've also seen suggestions to mix some cement and sand, brush it dry into all the cracks, and spray it with a light mist of water, but that would make any future repairs very difficult.