In the "Doctor vs. Classic Strange" fight, I said (I think rightly) that it would depend on prep. With prep, Strange's feats were better. Sans prep, the Doctor could do more in a hurry, keep the pressure up, and likely win. Not that Strange was powerless sans prep, but the Doctor's thought-based magic is more suited for non-prep than Strange's methods.
Fate isn't Strange, and he also doesn't have quite the power gap when you give him prep. If the helmet is on and the bearer knows his ****, it's pretty much on in any scenario. So it's a tough fight to call.
Sorry, I'm not offering a verdict here. I really don't know.
Hectors hard application of magic matches those of the Doctor.
The Last known Doctor would be Swift. And it doesn't matter if its Swift, Habib, or Jeroen. It takes a combination of skill, and mental resolution to defeat Hector.
Swift being the Doctor isn't an issue. They all inherit the experiences of those before them. She'd have access to Habib and Jeroen's feats.
You'd have to define "manipulating." But all of his relevant feats are in his respect thread. He's got high-level matter manip, time manip., telekinetic, and mental powers. He thinks something and it happens. It's not reality manip, as was established definitively on a couple occasions. But he doesn't have to, for example, know what he's doing like Firestorm or something, who had to know the formulas for his transmutation powers.
It's often described as magic, but is actually very grounded in how it works compared to the "deus ex" qualities of most magic.
He does have limits, of course. But his powers are very seamless. Without a phrase uttered or a movement, he could travel back in time to Fate's weakest point and flood him with the combined miseries of billions of people while simultaneously trying to transmute him into cupcakes and evaporate him with planet-destroying blasts. Three of four of those are actual feats of his; the other is well within his power.
Which is why Fate would need to be ready, and able to counter a LOT in a hurry. He's probably up to the task in his best incarnations, but it wouldn't be an easy fight by any stretch.
Technically he did. But it's a non-feat. A dude who was a human bomb was about to explode. The Doctor transported him to a dimension of nothingness where the guy became that universe's Big Bang (apparently the scale was different in that reality), thus creating a universe. It was a dimension-hopping feat, nothing more. The Doctor doesn't hold the power of a Big Bang in his fingertips, sadly.