I guess that the answer to your first question is *no*, based on the following: If the union of the $G_i$ were dense in $G=\mathrm{Diff}(M)$, then, presumably, for $i$ sufficiently large, the action of $G_i$ would be primitive (i.e., it would not preserve any nontrivial foliation) and locally transitive. The list of the effective, primitive, transitive Lie group actions is known, and, by examining this list, one sees that the dimension of such a group acting on an $n$-manifold is at most $n^2{+}2n$.

The answer to your second question depends on the manifold. First, I suppose you have to restrict to the case in which $M$ actually *has* a transitive smooth action of a finite dimensional Lie group. (For example, any compact orientable surface $M^2$ of genus $2$ or more is not a homogeneous space.)

It is not hard to come up with cases for which there is no upper bound. For example, let $M = \mathbb{R}^2$ and consider the group $G_d$ that consists of transformations of the form $\phi(x,y) = \bigl(x{+}a,\ y{+}p(x)\bigr)$ where $a$ is any constant and $p$ is any polynomial of degree $d$ or less. Then $G_d$ acts transitively on $M$ for all $d\ge0$ while $\dim G_d = d+2$. Thus, for $M=\mathbb{R}^2$ the dimension of such $H$ can be arbitrarily high. A similar argument with trig polynomials will provide such an example on the torus, which is compact. (N.B.: The action of $G_d$ is not primitive since it preserves the foliation by the lines $x=c$; thus, this example does not contradict my first paragraph.)

On the other hand, for $M=S^2$, there is an upper bound for the dimension of a connected Lie group that acts faithfully and transitively on $M$. That upper bound is 8 ($=2^2+2\cdot2$) and is achieved by $SL(3,\mathbb{R})$ acting on $S^2$ regarded as the space of oriented lines in $\mathbb{R}^3$. In fact, you can say more: Any connected, transitive finite dimensional Lie subgroup of the diffeomorphism group of $S^2$ is conjugate to one of $SO(3)$, $PSL(2,C)$, or $SL(3,\mathbb{R})$. The latter two are maximal and contain the first one as maximal compact. (The easiest proof that I know of these statements uses the classification of primitive actions, at least in dimension $2$.)