Yes. Let $H$ be a finitely generated subgroup of $F$. Let $[0,a]$ be the largest interval where all elements of $H$ are equal to the identity. Then consider the map that sends $h\in H$ to the $\log_2$ of the (right) slope of $h$ at $a$. This map is a non-trivial homomorphism of $H$ into $\mathbb{Z}$.

A more fancy (but essentialy the same) proof is this: by Thurston's stability theorem (W. Thurston. A generalization of the Reeb stability theorem. Topology 13 (1974), 347- 352), every group of $C^1$ diffeomorphisms of the closed interval is locally indicable, and by Ghys and Sergiescu, $F$ is a subgroup of the group of $C^\infty$ diffeomorphisms of the interval $[0,1]$ (see E. Ghys and V. Sergiescu: Sur un groupe remarquable de diffeomorphisms du cercle, Comm. Math. Helv. 62 (1987) 185–239.).

Yes. Let $H$ be a finitely generated subgroup of $F$. Let $[0,a]$ be the largest interval where all elements of $H$ are equal to the identity. Then consider the map that sends $h\in H$ to the $\log_2$ of the (right) slope of $h$ at $a$. This map is a non-trivial homomorphism of $H$ into $\mathbb{Z}$.

A more fancy (but essentialy the same) proof is this: by Thurston's stability theorem (W. Thurston. A generalization of the Reeb stability theorem. Topology 13 (1974), 347- 352), every group of $C^1$ diffeomorphisms of the closed interval is locally indicable, and by Ghys and Sergiescu, $F$ is a subgroup of the group of $C^\infty$ diffeomorphisms of the interval $[0,1]$ (see E. Ghys and V. Sergiescu: Sur un groupe remarquable de diffeomorphisms du cercle, Comm. Math. Helv. 62 (1987) 185–239, an easy proof using diagram groups can be found in V.S. Guba and M. Sapir. Diagram groups are totally orderable. J. Pure Appl. Algebra, 205(1):48–73, 2006).

Yes. Let $H$ be a finitely generated subgroup of $F$. Let $[0,a]$ be the largest interval where all elements of $H$ are equal to the identity. Then consider the map that sends $h\in H$ to the $\log_2$ of the (right) slope of $h$ at $a$. This map is a non-trivial homomorphism of $H$ into $\mathbb{Z}$.

Yes. Let $H$ be a finitely generated subgroup of $F$. Let $[0,a]$ be the largest interval where all elements of $H$ are equal to the identity. Then consider the map that sends $h\in H$ to the $\log_2$ of the (right) slope of $h$ at $a$. This map is a non-trivial homomorphism of $H$ into $\mathbb{Z}$.

A more fancy (but essentialy the same) proof is this: by Thurston's stability theorem (W. Thurston. A generalization of the Reeb stability theorem. Topology 13 (1974), 347- 352), every group of $C^1$ diffeomorphisms of the closed interval is locally indicable, and by Ghys and Sergiescu, $F$ is a subgroup of the group of $C^\infty$ diffeomorphisms of the interval $[0,1]$ (see E. Ghys and V. Sergiescu: Sur un groupe remarquable de diffeomorphisms du cercle, Comm. Math. Helv. 62 (1987) 185–239.).