Does there exist a set $F$ of monotone continuous functions $f \colon [0,1] \to [0,1]$ with the following properties?

1. For each $f \in F$ there exists $x \in [0,1]$ such that $f(x)=1$.
2. There exist $0<m<M<\infty$ such that for all $x \in [0,1)$ and $f \in F$, if $f(x)<1$ then $f$ is $C^2$ at $x$ with $$ m \ < \ f'(x),f''(x) \ < \ M. $$
3. There exists $c \in (0,1)$ such that for all $x \in [0,1)$ and $f \in F$ with $f(x)<1$, $$ x < c \ \Longleftrightarrow \ f'(x)<1. $$
4. The semigroup $S_F := \{f_j \circ \ldots \circ f_1 \, : \, j \geq 1, \, f_1,\ldots,f_j \in F \}$ contains functions $\{g_n\}_{n \in \mathbb{N}}$ such that as $n \to \infty$, $g_n$ converges uniformly to the identity function.

Does there exist a set $F$ of monotone continuous functions $f \colon [0,1] \to [0,1]$ with the following properties?

1. For each $f \in F$ there exists $x \in [0,1]$ such that $f(x)=1$.
2. There exist $0<m<M<\infty$ such that for all $x \in [0,1)$ and $f \in F$, if $f(x)<1$ then $f$ is $C^2$ at $x$ with $$ m \ < \ f'(x), \! f''(x) \ < \ M. $$
3. There exists $c \in (0,1)$ such that for all $x \in [0,1)$ and $f \in F$ with $f(x)<1$, $$ x < c \ \Longleftrightarrow \ f'(x)<1. $$
4. The semigroup $S_F := \{f_j \circ \ldots \circ f_1 \, : \, j \geq 1, \, f_1,\ldots,f_j \in F \}$ contains functions $g_1,g_2,g_3,\ldots$ converging uniformly to the identity function.