1. Should I completely avoid the use of those methods?
Yes, you need to avoid using those methods for application code.
2. The methods have been defined for a reason. So when should I use them? In which context? For what purposes?
I consider it as an API design accident. Somewhat forced by composite components having special ideas about child sizes. "Imperceptibly", because they should have fulfilled their needs with a custom LayoutManager.
3. What exactly are the negative consequences of using those methods? (I can only think to add portability between systems with different screen resolution.)
Some (unfinished, and regrettably the links are broken due to the movement of SwingLabs to java.net) technical ideas are for instance specified in the Rules (hehe). Ethically, posing tonnes of work onto your impoverished fellow who has to maintain the code and has to track down a broken layout.
4. I don't think any LayoutManager can exactly satisfy all desired layout needs. Do I need to implement a new LayoutManager for every little variation on my layout?
Yes, there are LayoutManagers sturdy enough to provide a very good approximation to "all layout requirements". The main three are JGoodies FormLayout, MigLayout, DesignGridLayout. So no, in tradition, you infrequently write LayoutManagers except for simple extremely specialized environments.
5. If the answer to 4 is "yes", won't this lead to a proliferation of LayoutManager classes which will become difficult to maintain?
The answer to question 4 is "No"
6. In a situation where I need to define proportions between children of a Component (for example, child 1 should use 10% of space, child 2 40%, child 3 50%), is it possible to achieve that without implementing a custom LayoutManager?
Either of the Big-Three can, I never tried GridBag though (too much trouble for too little power).