As for question (1): assuming cocomplete cartesian monoidal category means that cartesian products distribute over colimits (and I know from past experience that you do mean this, Martin), then it's true that the monad $T$ preserves reflexive coequalizers. The main thing you need is that finitary power functors $c \mapsto c^n$ preserve reflexive coequalizers. This is a corollary of a result that you can find on the first page of chapter 0 of Johnstone's Topos Theory, which is a $3 \times 3$ lemma stating that if the rows and columns are reflexive coequalizer diagrams, then so is the diagonal. It easily follows from this lemma that for example the squaring functor $c \mapsto c \times c$ preserves reflexive coequalizers, and a similar inductive argument allows you to extend this to any finite power $c \mapsto c^n$.
To derive the fact that monads $T: C \to C$ based on a Lawvere algebraic theory $\theta$ preserve reflexive coequalizers, write
$$T(c) = \int^{n \in \mathrm{FinSet}^{op}} \hom_\theta(i(n), i(1)) \cdot c^n$$
where the tensor $S \cdot c$ of a set $S$ with an object $c$ is the coproduct of copies of $c$ indexed over $S$. (Here $i: \mathrm{FinSet}^{op} \to \theta$ denotes the unique (up to isomorphism) map of Lawvere algebraic theories, viewing $\mathrm{FinSet}^{op}$ as the "initial" Lawvere algebraic theory.) Since coend functors and tensor functors $S \cdot -$ preserve reflexive coequalizers, as does $c \mapsto c^{n}$, we see that $T$ does as well.
I can't think of a more direct nice description of the composite left adjoint $F$, nor do I think one is needed because I think the description you gave is plenty nice.
As for (2): the underlying functor is definitely not monadic. It's not even a right adjoint, because for example for $C = \mathrm{Vect}_k$, it fails to preserve the terminal object (which in $\mathrm{AbHopf}(\mathrm{Vect})_k$ is the monoidal unit $k$, as is the case just in $\mathrm{CoMon}(\mathrm{Vect})_k$).
Edit: Since this came up in comments, let me provide an alternative proof of the fact that finite power functors on a cocomplete cartesian monoidal category preserve reflexive coequalizers. Recall that a category $J$ is sifted if the diagonal functor $J \to J \times J$ is final (result due to Gabriel and Ulmer). A prototypical example is where $J$ is the generic parallel pair equipped with a section in common. Then follow Steve Lack's soft proof here, which uses just the assumption that $C$ is cocomplete cartesian monoidal and the finality of the diagonal on $J$, to show the binary product $C^2 \to C$ preserves reflexive coequalizers. Similarly, the $n$-fold product $C^n \to C$ preserves reflexive coequalizers. The $n$-fold power on $C$ is a composite of the diagonal $\Delta: C \to C^n$ (which is a left adjoint, thus colimit-preserving) with the $n$-fold product, so it too preserves reflexive coequalizers.