"Fall", "fell", "felled"

Solution 1:

Your only error is that the causative form is fell in the present tense: A logger fells trees today. Otherwise you've got it right.

causative: fell, felled, has/be felled, as opposed to
intransitive: fall, fell, has fallen

However, felling a dynasty or regime, or anything except a man, animal, or tree, is pretty rare today; OED 1 was already marking it as obsolete in 1895. Topple (in the transitive use) is more common.

EDIT -- taking a healthy bite of my words.
On review I find that falling a tree is in fact still in use. The usage appears, on a quick Google scan, to be confined to the western US, and to non-formal usage; but it's something more than rare.

As for declension: Google reports 5 (non-duplicate) instances of falled a tree, 17 of he fell a tree, 13 of has fallen a tree, and none of has falled a tree. So the regular strong form seems to prevail.

Solution 2:

These are two completely different verbs. There's

fall | fell | fallen

which is intransitive. Also, there's

fell | felled | felled

which is transitive.

So a tree can fall. A tree fell. Ten trees have fallen today.

John can fell a tree. John felled a tree. John has felled ten trees today.

Your sentence fragment "a logger falls trees" is incorrect, as far as I know.