Regularity of the heat kernel Let $(M,g)$ be a compact Riemannian manifold. 
Let $H:M\times M\times\mathbb{R}_{>0}\to\mathbb{R}$ be the heat kernel. i.e. $H\in C^0(M\times M\times\mathbb{R}_{>0})$ is the unique continuous function such that for all $y\in M$, 
(A) $H^y\in C^{2,1}(M\times\mathbb{R}_{>0})$
(B) $\left(\Delta^g-\dfrac{\partial}{\partial t}\right)H^y=0$
(C) $\displaystyle\lim_{t\to0}H^y_t=\delta_y$
, where $H^y(x,t):=H(x,y,t)$, $H^y_t(x):=H^y(x,t)$, 
$C^{2,1}(M\times\mathbb{R}_{>0}):=\{\varphi:M\times\mathbb{R}_{>0}\to\mathbb{R}|\text{ For each chart }(U;x^1,\cdots,x^m)\subset M, \dfrac{\partial\varphi}{\partial t}, \dfrac{\partial\varphi}{\partial x^i},\text{ and }\dfrac{\partial^2\varphi}{\partial x^i\partial x^j}:U\times\mathbb{R}_{>0}\to\mathbb{R} \text{ are well defined and continuous.}\}$. 
${\bf [Question 1]}$ From (A) and (B) above it is derived that $H^y\in C^\infty(M\times\mathbb{R}_{>0})$ for all $y\in M$. How about the regularity of H as a function on $M\times M\times\mathbb{R}_{>0}$? Does it hold that $H\in C^\infty(M\times M\times \mathbb{R}_{>0})$? If not, aren't there any regularity result of $H:M\times M\times\mathbb{R}_{>0}\to\mathbb{R}$ which is useful to exchange integrals and differentiation?
${\bf [Question 2]}$ Suppose that $F:M\times[0,T]\to \mathbb{R}$ is a continuous function. Then is it true that the function
\begin{eqnarray}
u(x,t):=-\int_0^t\int_M H(x,y,t-\tau)F(y,\tau)\mu_g(dy)d\tau
\end{eqnarray}
belongs to $C^{1,0}(M\times[0,T])\cap C^{2,1}(M\times(0,T))$? 
${\bf [Question 3]}$
Suppose that $f:M\to\mathbb{R}$ be a $C^1$ function. Does the function
\begin{eqnarray}
v(x,t):=\int_M H(x,y,t)f(y)\mu_g dy
\end{eqnarray}
belong to $C^{1,0}(M\times[0,\infty))\cap C^{2,1}(M\times\mathbb{R}_{>0})$? 
Please tell me also references. Thank you. 
 A: For question 1, see Theorem 5.2.1 of E. B. Davies's book Heat Kernels and Spectral Theory, which asserts that indeed the heat kernel is a $C^\infty$ function on $M \times M \times (0,\infty)$.  (It also applies to Riemannian manifolds $M$ which are complete but not compact.)
For question 2, it suffices to assume $F$ is bounded and measurable.  Differentiation under the integral sign will show that $u \in C^\infty(M \times (0,T))$.  Indeed, for this it suffices to assume $F$ is bounded and measurable.  To get continuity up to $t=T$, extend $F$ to $\tilde{F} : M \times [0,T+\epsilon]$ by $\tilde{F}(x,t) = F(x,t)$ for $t \le T$ and $\tilde{F}(x,t) = 0$ for $t > T$.   Then the corresponding function $\tilde{u}$ is continuous (even smooth) on $M \times (0, T+\epsilon)$ and $\tilde{u}(x,t) = u(x,t)$ for $t \le T$.  
To get continuity at $t=0$, simply note that
$$|u(x,t)| \le t \cdot\left(\sup_{M \times [0,t]} |F|\right)\left( \sup_{\tau \in [0,t]} \int_M H(x,y,t-\tau) \mu_g(dy)\right)$$
But $\sup_{M \times [0,t]} |F|$ is finite if $F$ is bounded, and $\int_M H(x,y,t-\tau) \mu_g(dy) = 1$ for any $x, t, \tau$.
I guess you still want to show that the spatial derivatives of $u$ are continuous up to $t=0$.  That shouldn't be hard but maybe takes a little more thought.
For question 3, as before, if $f$ is merely bounded and measurable then $u \in C^\infty(M \times (0,\infty))$ by differentiating under the integral sign. 
Still working on continuity at 0.  It's really just going to come from the continuity of $f$, the fact that $H(x,y,t) \to \delta_x$ and the triangle inequality.
A: Smoothness in all the questions has a local character. And in local coordinates it's a parabolic equation $Lu=F$ with (smooth) variable coefficients. So the local theory in $\mathbb R^n$ will do.
On Question2 the answer is no. If function $F$ is continuous it does not follow that $u$ is locally from $C^{2,1}$. For the heat equation see Nonclassical solution to u_t-\Delta u=f in one space dimension? question here.
On Question3 the answer is yes. As it was mentioned it's a question of local regularity. And for $\mathbb R^n$ one can differentiate the equation $Lu=0$ with respect to the space variable $x_i$, transfer all the terms with $u$ to the rhs and obtain a Cauchy problem for $\partial_iu$ with a continuous initial condition $\partial_i f$ and a continuous rhs. It is included in the definition of a fundamental solution that $v(x,t)\to f(x)$ as $t\to0\!+$ for continuous $f$, see, for example, A. Friedman, Partial Differential Equations of Parabolic Type.
