Some thoughts, regarding question (a): 

In the case of a pointed, cocommutative hopf algebra $H$ over a field $k$ of characteristic $0$, by the Cartier-Konstant-Milnor-Moore theorem (see:  https://mathoverflow.net/questions/255767/classification-of-quasitriangular-hopf-algebras) we have that 
$$
H\cong U\big(P(H)\big)\sharp kG(H)
$$
where $P(H)$ is the lie algebra of the primitives, $U(.)$ its UEA and $G(H)$ the group of the group-likes. 
So, it turns, that a general condition for $\phi$ to be surjective, is the set of grouplikes to be trivial, in which case $H$ is generated by its primitives and $H\cong U\big(P(H)\big)$.  
(this includes as a special case the cocommutative hopf algebras over algebraically closed fields, since these are pointed). 

A similar result -again refering to a case with trivial grouplikes- is the situation of a cocommutative, irreducible hopf algebra $H$ over a field $k$ of characteristic zero: then we have the hopf algebra isomorphism 
$$
H\cong U\big(P(H\big))
$$