The scene of him running down the tower in 3 gives some indication of hos speed, from there he gets faster
Yeah, but he couldn't outrun his own bullet. Eventually the bullet hit the sword and Dante was able to catch up to the sword, but this is likely because the sword itself was slowed down due to drag and hitting multiple blood-goyles who are light enough to fly. Still fast though.
Alastor imbues him with lightnig speed in dmc1
I do know that the Alastor gives it's user electric properties, but are there feats that support he could move at lightning speeds? Certainly Dante's ability to fire lightning out of his hands should be faster than bullets. One of Beowulf's attack descriptions has Dante making small super novas with his hands, though I doubt the actual power is comparable to that of an actual supernova.
He can also keep up with and best blitzes with ease which are lightning speed
From what I saw with the cutscene, Dante missed his first attack against the blitz, and seeing as how trying to physically hit it with a melee weapon damages the player, it is implied that Dante had to use bullets against the demon. The fact that Dante has sped up his bullet projectiles with his demon powers is a possibility so there's that.
The savior scene in 4 where dante is hanging off his sword stuck on its hand is a clue too , it slams its hands together really fast
Dante was able to avoid these true, but was it really necessary for Dante to be at bullet speeds to do so?
Plus he can teleport and dash insanely fast with trickster
Now Dante teleporting and dashing might make him faster than bullets, the only thing is it is hard to measure but I can believe that. After all if he can dash in midair then it's possible he can make himself even faster, if not in specific directions.