Pino,
I'm very far from a master in grafting figs. Just started 3 years ago with figs and my first attempts all failed (wrong grafting methods that were not the best for figs, like bark graft, or not enough protection against moisture loss).
I began again and this time i tried chip grafting and, ignoring some recommendations, whip and tongue and the results were way better.
In the last couple of years i have done 34 whip and tongue grafts, 29 of those were successful. In 3 of the one's that weren't the scion was in bad shape (they were last attempts in trying to save the upper part of a rotted scion that had failed rooting), the other 2 were thin curved cuttings and i did a poor job of aligning the cambium layers, so i consider this method the most successful i use). The problem with this method is finding the same size rootstock to graft.
In the same period i have done almost twice as much chip-buds (roughly 56) and the success rate is lower (some 34 took, although a few more could still break bud when spring comes - i had some chips that stay dormant for a full year).
I have to say that most of those who failed were experiments, when i was trying to see the limits of the method.
This October, i have done a few more late fall chip-buds (around 16 chips from 4 cuttings), that looked good 2-3 weeks after grafting. I am not counting those as i will have to wait they leave dormancy to see if they succeeded, but i am confident.
To answer your questions:
1. I don't try to rehydrate the cuttings. In rooting that can probably help, but in grafting i believe it's the sap from the rootstock that is going to do much of the work (in providing the energy for both cambium tissues to join and callus, as long as both tissues are alive and in good condition), so i take the dormant scions out of the fridge and graft them immediately.
2. I try to graft in one year wood, when it's possible, and the scion is actively growing. When i make the grafting cut in the rootstock i try to determine if the flow of sap is strong, medium or weak (it depends on time of year and vigor of the rootstock) and accordingly i make more or less incisions (shallower or deeper as needed), usually near the base of the branch being grafted. I don't do the incisions hours or days before the graft. Nevertheless, if i see an unusually strong flow, i may postpone the graft - will do others first, to let the sap flow freely for a bit, on that particular branch.
3. I've seen good callus develop during most of the grafting season (from spring to fall), but the rapid growth of callus tissue in spring helps a lot, so it can help to cover some mistakes that may cause the graft to fail, in not so benign seasons. In the fall you can have similar situations (during those warmers days that resemble spring) but the problem is the possibility of cold spells that make everything stop and that may jeopardise the survival rate of undeveloped grafts in winter.
4. To me, the problem with these tools is that they are not very flexible. if the scion is not 100% straight, or you place it not exactly the same way as the rootstock, as Cheryl is saying, there is a higher probability of misaligning the cambium layers. When doing the grafting cuts and the alignment "manually" we can compensate better for different diameters or curved scions and make the cambiums cross at some point what virtually guarantees success, if everything else is correct.
Cheryl
I prefer that my rootstock is actively growing when grafting. If both are dormant, there is a chance that the scion "wakes up" first and, without an established connection to the rootstock, it will dry up and die.
Patch bud grafting has some pros and cons. I think that, when done correctly, it's one of the best methods to graft to older wood and very different diameters are not a problem. It provides for strong unions and rapid growth. But it only works if the bark is slipping and it's not very easy to do. With fig scions particularly, removing a patch with a bud can be quite difficult, particularly in some varieties that have a tendency to have very short internodes, and, usually, the fig buds have 3 "anchors" points (probably clusters of meristematic cells that should go with the patch but stubbornly want to stay in the scion), and these can cause the patch to break, if you are not very careful removing it.

Also, for the "window" on the rootstock and the patch to align perfectly it's better to use a double grafting knife that does the 2 cuts exactly the same size (and they are not very common tools). Nevertheless, i don't have enough experience with this method with fig trees to give advices (i've only done 5 and 2 failed). I might try it more in the future but didn't feel the need to during last year.
So, in short, i think that chip budding is much simpler to do and it provides good enough results.