Is regeneration a naturally biological trait of the timelords, or is it a result of timelord science? I've seen conflicting stories, especially with the ability to give new cycles and force regeneration it does seem to be more of an unnatural ability but I'm unclear.
Regeneration is one of those aspect about Doctor Who that isn't entirely clear on its own rules, and isn't always portrayed consistently from episode to episode. (Other similar aspects include the rest of Doctor Who).
However, to the best of our knowledge, the ability to regenerate upon near-death is a Time Lord capability that is bestowed on new Time Lords by their high Council. The basic idea is that Time Lords are all from the race of people called Gallifreyans, but not all Gallifreyans are Time Lords. At some point in their past, Rassilon (and some others, but he's the one that's usually name-dropped) developed all kinds of awesome technology that let to the creation of the Time Lords.
One of those things was the ability to regenerate, but the process was strictly regulated: each Time Lord got 13 of them, after which they died for good.
The primary reason we have to believe this is, as you pointed out, because the High Council has been shown, on more than once occasion, to gift bonus regenerations to Time Lords as reward for their service. For example, they once offered to give The Master a new set of regenerations for helping The Doctor; they also resurrected him to fight in the Time War, and he's regenerated at least twice since then.
They've also been shown to be able to force a regeneration even when the Time Lord was in perfect health, as they did with the Second Doctor into the Third.
Finally, in the mini-episode "Night of the Doctor", The Eighth Doctor lands on Karn, which is inhabited by Gallifreyan exiles called the Sisterhood of Karn, and dies in a crash. The Sisterhood revive him, and tell him they have the ability to control his regeneration because, in their words:
Time Lord science is elevated on Karn.
All of this strongly implies that the ability to regenerate is a scientific/technological advancement given to the Time Lords, and not an ability they naturally developed.