Let $d\in\mathbb N$, $k\in\{1,\ldots,d\}$, $\alpha\in\mathbb N$, $M$ be a $k$-dimensional embedded $C^\alpha$-submanifold of $\mathbb R^d$ with boundary, $\nu_{\partial M}$ denote the outward pointing unit normal field on $\partial M$ and $\operatorname P_M(x)$ denote the orthogonal projection of $\mathbb R^d$ onto the tangent space $T_x\:M$ of $M$ at $x\in M$.
Question 1: I would like to show that $M\ni x\mapsto\operatorname P_M(x)$ is $C^{\alpha-1}$-differentiable. Moreover, I would like to find a formula for the pushforward $T_x(\operatorname P_M)$, $x\in M$.
Question 2: Moreover, I would like to understand the concept of the tangential gradient. To be precise, if $E$ is a $\mathbb R$-Banach space (assume $E=\mathbb R$ if it's easier to follow for you) and $f:\partial M\to E$ is $C^1$-differentiable, isn't the tangential differential (or "gradient", if $E=\mathbb R$) of $f$ at $x\in\partial M$ precisely equal to $${\rm D}_{\partial M}f(x):={\rm D}f(x)\circ\operatorname P_{\partial M}(x)\tag1,$$ where ${\rm D}f(x)=T_x(f):T_x\:\partial M\to T_{f(x)}f(\partial M)$ denotes the pushforward of $f$ at $x$?
Regarding question 1: It's sufficient to consider the dependence on $x\in M$ of $\operatorname P_M(x)$ locally. So, let $\Omega$ be an open subset of $M$ and $\phi$ be a $C^\alpha$-diffeomorphism from $\Omega$ onto $\mathbb H^k:=\mathbb R^{k-1}\times[0,\infty)$. Note that $$T_x\:M=T_x(\phi)^{-1}\mathbb R^k\;\;\;\text{for all }x\in\Omega\tag2.$$ Let $(e_1,\ldots,e_k)$ denote the standard basis of $\mathbb R^k$. By $(5)$, $$\sigma_i(x):=T_x(\phi)^{-1}e_i\;\;\;\text{for }i\in\{1,\ldots,k\}$$ is a basis of $T_x\:M$ for all $x\in\Omega$. Since $\phi$ is a $C^\alpha$-diffeomorphism, $$\Omega\ni x\mapsto T_x(\phi)^{-1}=\left({\rm D}\phi^{-1}\circ\phi\right)(x)\tag3$$ is $C^{\alpha-1}$-differentiable and hence $\sigma_1,\ldots,\sigma_k$ are $C^{\alpha-1}$-differentiable. Now let $(\tau_1(x),\ldots,\tau_k(x))$ denote the orthonormal basis of $T_x\:M$ obtaine from $(\sigma_1(x),\ldots,\sigma_k(x))$ by the Gram-Schmidt orthonormalization process for $x\in\Omega$. Noting that this process does not decrease regularity and noting that$^1$ $$\left.\operatorname P_M\right|_\Omega=\sum_{i=1}^k\tau_i\otimes\tau_i\tag4,$$ we are done. Or did I made a mistake at some point? And how can we compute $T_x(\operatorname P_M)$, $x\in M$?
Regarding question 2: Note that $$\nu_{\partial M}(x)\in T_x\:M\cap N_x\:\partial M\tag5\;\;\;\text{for all }x\in\partial M$$ and, by definition, the tangential differential of $f$ at $x\in\partial M$ in direction $v\in T_x\:M$ should be given by $$T_x(f)\left(v-\langle v,\nu_{\partial M}(x)\rangle\nu_{\partial M}(x)\right),\tag6$$ but in the special case $k=d$, where $T_x\:M=\mathbb R^d$. Now, if $(\Omega,\phi)$ and $(\tau_1(x),\ldots,\tau_k(x))$ are as above, $$T_x\:\partial M=T_x(\phi)^{-1}\partial\mathbb H^k\tag7$$ and hence $(\tau_1(x),\ldots,\tau_{k-1}(x))$ is an orthonormal basis of $T_x\:\partial M$ for all $x\in\partial\Omega$. Let $x\in\partial\Omega$. Since $\nu_{\partial M}(x)\in N_x\:\partial M=\left(T_x\:\partial M\right)^\perp$, $$\langle\nu_{\partial M}(x),\tau_i(x)\rangle=0\;\;\;\text{for all }i\in\{1,\ldots,k-1\}\tag8$$ and it should follow that $(\tau_1(x),\ldots,\tau_{k-1}(x),\nu_{\partial M}(x))$ is an orthonormal basis of $T_x\:M$. Moreover, for what it's worth, it should hold $$\tau_k(x)=\frac{\nu_{\partial M}(x)}{\left|\langle\nu_{\partial M}(x),\tau_k(x)\rangle\right|^2}\tag9.$$ In particular, it should hold $$\operatorname P_{\partial M}=\left.\operatorname P_M\right|_{\partial M}-\nu_{\partial M}\otimes\nu_{\partial M}\tag{10},$$ i.e. $\operatorname P_M(x)-\nu_{\partial M}(x)\otimes\nu_{\partial M}(x)$ should be the orthogonal projection of $\mathbb R^d$ onto $T_x\:\partial M$ and the tangential differential of $f$ at $x$ should be equal to $(1)$. Did I made any mistake at some point?
$^1$ As usual, $u\otimes v:=\langle\;\cdot\;,u\rangle v$ for $u,v\in\mathbb R^d$.