Let $c:S^1\rightarrow\bf{R}^2$ be a closed curve (i.e. $c$ is injective). Does there exist a homotopy $h_c:S^1\times[0,1]\rightarrow\bf{R}^2$ between $\sf{id}_{S^1}$ and $c$ such that $p\mapsto h_c(p,t)$ is injective for $t\in[0,1]$?
If yes, is it also possible to have $h_c$ smooth on $S^1\times[0,1)$?
I don't believe this is true. Take $c=-id_{S^1}$, that is $S^1$ with the opposite orientation, and assume that $h:S^1\times I\rightarrow \mathbb{R}^2$ is a homotopy $id_{S^1}\simeq -id_{S^1}$ through simple closed curves. Then at each time $t\in I$ the simple closed curve $h_t:S^1\rightarrow \mathbb{R}^2$ has a well defined degree of $deg(h_t)=\pm 1$ which by continuity must be the same for all values of $t\in I$. Since $deg(h_0)=deg(id_{S^1})=1$ and $deg(h_1)=deg(-id_{S^1})=-1$ we have a contradiction so $h$ cannot exist.