Kernel for the Hodge chain homotopy

Question

Let $M$ be a Riemannian manifold of dimension $n$. For $p\in \mathbb Z$, let

1. $C^p=A^p(M;\mathbb R)$,
2. $d:C^p\to C^{p+1}$ be the differential,
3. $*:C^p\to C^{n-p}$ be the Hodge star, satisfying $**=(-1)^{p(n-p)}$,
4. $\delta:C^p\to C^{p-1}$ be the formal adjoint of $d$, given by $\delta=(-1)^{n(p+1)+1}*d*$,
5. $\Delta:C^p\to C^p$ be the Laplace Beltrami operator, given by $\Delta =d\delta+\delta d$,
6. $(,):C^p\otimes C^p\to \mathbb R$ be the Hodge inner product, given by $(\alpha,\beta)=\int_M\alpha\wedge *\beta$,
7. $H^p=\ker\Delta\subset C^p$ be the space of harmonic differential forms,
8. $i:H^p\to C^p$ be the inclusion.

Hodge theorem states that $C^p=\Delta(C^p)\oplus H^p$ with respect to the inner product $(,)$. Let

1. $H:C^p\to H^p$ be the orthogonal projection
2. $G:C^p\to C^p$ be the unique operator satisfying $\Delta \circ G(\alpha)=\alpha-i\circ H(\alpha),$
3. $h:C^p\to C^{p-1}$ be the homotopy equivalence $h=\delta \circ G$.

One can easily check that $$ d\circ h+h\circ d=1-i\circ H. $$ In fact, this is generally carried out in the setting where $C^p$ is a Sobolev completion of $A^p(M;\mathbb R)$, or $L^2$ differential forms. In this case, $G$ increases regularity by $2$.

My questions are:

1. Does $h$ increase regularity by $1$?
2. Can we write $(h\alpha,\beta)$ as an integral $$ (h\alpha,\beta)=\int_{M\times M}k(x,y)\alpha(x)\beta(y)d\text{Vol}(x)d\text{Vol}(y), $$ for some $k$, and what is the regularity of $k$? Is it merely a tempered distribution, or is it actually an $L^2$ function on $M\times M$?
3. How does all this relate to the heat kernel $e^{-t\Delta}$ obtained from the functional calculus for $\Delta$? (or for the Dirac operator $D:C\to C$ satisfying $D^2=\Delta$)