Warning: I seem to differ from sinker on a few points:
I have Conviction of +3 and Discipline of +4. This means I can put no more than 3 shifts of power into casting per turn, right?
You could opt to put in more power on a given exchange, but power over your Conviction costs Mental Stress. So if you opted to front-load your ritual (a good idea for some rituals), you could op to put, say, 5 shifts into your first exchange: it would cost you 2 Mental Stress, and you would need to roll 5 shifts on Discipline to control it, but if you succeed, then you can squeeze the remaining shifts out 1 at a time. Some thaumaturgists will opt to build up as much as they safely can in the first exchange so that they only have to trickle in the remaining shifts, thus making it less likely they will lose control and suddenly have to deal with almost the entire ritual's collection of power as backlash. Because if you fail a roll, you take ALL of the bucket as stress.
And I roll each turn at a +4? So I'd have to roll at a +3 difficulty the first two times, and then the third time at a +2, to get the spell to go off properly in the minimal time. Correct so far?
Yes, ideally.
Okay, so let's say I roll a total +2 the first time. I can take Fallout, which means that the whole spell fails, or take 1 Mental stress as Backlash to keep going. Yes? Okay, let's say I do that.
No, you have to absorb the current bucket of shifts as stress if you want to keep the spell going, which is why it is good to frontload the ritual with shifts and go all-out to control it, so that you're less likely to lose the bucket on a later roll. You could theoretically take a 2-point Consequence and add that to the control roll, and then you wouldn't have technically lost control. You could also spend a Fate Point to tag an Aspect to add to the control roll.
But if your control roll fails for whatever reason, ALL of the accumulated shifts (in the case of round 1, that would be 3 shifts) go back to you as either Fallout or Backlash: not just the single shift you missed it by. It sucks, certainly, though 3 shifts is not the worst I've seen.
Also, Backlash can be taken as either Physical Stress or Mental Stress (but not split between them).
The second time, I get a +5. Whee! So what happens to the extra 2 shifts in this case? Are they just gone? Or can I put them towards future power of the spell?
Nothing. They go away. Overkill. Your GM may opt to allow you to generate "spin" (see Your Story for those optional rules) but those are mostly used in attack/defense situations.
The third time, I only need a +2, but I end up with a +1 (stupid dice). This means I have to take the extra 1 Mental stress or lose the spell. Now, is it possible I could roll over my extra +2 from the second roll to prevent the backlash? Or do I have to reach each roll independently? Is there any way to get extra shifts of power at this point, like I can with Evocation? Or am I pretty much stuck with 8 or nothing?
Each roll is independent. You could theoretically use Fate Points to tag an Aspect to help with a given roll. Your GM may also allow you to have placed one or more navel-gazing maneuvers like "Supreme Concentration" before the ritual, and instead of using them to make the initial Complexity, saving them up to free-tag in case rolls go wrong. I have also allowed people to take Consequences to boost rolls. I've also allowed Backlash Stress to be split between multiple casters.
But yes, you are stuck with 8 or nothing.
Okay, let's say I get exactly 8 and the spell hits. They then roll a Discipline defense at a +8 difficulty to resist the spell, right? And then, let's say they get +5. They then take 3 mental stress for the difference and then 8 more from the original power I put into the spell?
Yes, that sounds correct. Spells, and especially rituals, can be devastating if they are allowed to be completed. Nope, on further consideration, sinker is right on this one. My apologies.