1. As soon as the initial skinning operation is done, you'll want to move and test the rig to identify where any issues are. It's possible that some joints will be skinned perfectly fine without you needing to edit anything. From there, I personally tend to begin at the center of the character (spine, hips, shoulders, etc) and work my way outwards to the extremities. So your question is spot on, would recommend following your instinct!
2. As always, there are multiple ways to approach anything, so I'm just sharing my own process. I keep the rig in their default pose for the most part, but I do add temporary animation to any joint I am testing. For example, if I were painting the skin weights on the knee of a character, I would key the knee control at it's default position, then key the knee at a more extreme bend position, etc. This allows you to easily scrub in the timeline while you are painting skin weights, and stay in the tool, rather than needing to exit painting and select the control or joint. Once you are happy with a paint job, just return to default position and delete the keyframes. And again, your intuition is correct, in that you do want to test the joints in active/extreme positions rather than assuming it works!
3. Remember that the joints are part of a hierarchy. If you rotate the hip/thigh joint, then the knee and foot will follow right? If you rotate the knee joint, then the hip/thigh should stay stationary, but the foot will follow. (This is assuming forward kinematics and ignoring any IK). So if you were looking at, let's say.. the knee joint and painting skin weights, you are essentially telling the knee joint "hey, when you rotate, these vertices follow you", but also remember, that all of the joints down the hierarchy are already following the knee. What would happen if you told the knee joint to control all of the vertices from the knee area, down past the foot? When you go and bend the foot, nothing moves! Nothing will move because you given the knee full control and taken control away from the foot.
4. I'm not quite sure what your process is in relation to this question. In the paint weights tool, you can change things such as the paint operation, the opacity of your brush, as well as the value of the brush. While this may not be fully related to the issue you encountered, it should be noted that painting with a value of 1 will adding more control to your selected joint, and painting with a value of 0 will be removing control. Every vertex on your model that is currently skinned, wants/needs influence from a joint. So if you were to remove control/paint with a value of 0, the vertex you painted over will be assigned to the next logical joint. My advice here, is first - try to only paint with a value of 1 (control intensity with the opacity) and try to avoid painting with a value of 0. And secondly, remember that when you have a joint selected, you're only visually seeing that area of control, and all of the changes you make to that area do impact the nearby joints.
