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In the Highlander series, immortals have burned to death and recovered and fallen from great heights and recovered after healing. One had a hand cut off and it did not grow back, he put a metal hook thing in its place. Wounds on the throat don't seem to heal in both the movie and series. Also, immortals can apparently starve to death, revive and starve to death over and over.

1- So what is the "Quickening" in the first movie? Being "in tune" with the environment (flora and fauna) and tapping into the Earth's energies as well?
The Quickening is the source of their immortality, the sensation they feel when another immortal is near, and the energy they pass on when they are beheaded.
Also, the things you describe are implied several times. The Kurgan (and later MacLeod) having the ability to attract lightning, the scene with the deer, the birds scattering before the Kurgan arrives, and the brief but strange scene where MacLeod and Garfield are alone in the interrogation room.
And of course the Prize, which is MacLeod becoming "at one with all living things"; the same connection, but on a much grander scale.
It's a kind of magic
2- Have Immortals got extensive regeneration powers? What if they are burnt or they just jump from the 45th floor of a building?
They cannot die unless decapitated, and I think were supposed to assume that they can recover from anything until that happens. We see them take some big falls and recover after a few seconds, as well as getting shot repeatedly, stabbed and slashed without a mark on them. The only exception is the Kurgan's neck scar, but that was a near fatal wound.
There's even the magical side of it where Macleod not only survives drowning, but remains conscious and continues functioning without oxygen.
Their ability to regrow limbs and so forth is never addressed in the movie, but considering all the other injuries they sustain and recover from, it seems plausible.