Some thoughts, regarding question (a): 

In the case of a [pointed][1], 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 an [irreducible][2], cocommutative hopf algebra $H$ over a field $k$ of characteristic zero: then we have the hopf algebra isomorphism 
$$
H\cong U\big(P(H\big))
$$
If the above examples are of interest for your purposes, maybe the article: [Hopf algebras with one groupl-like element][3] and [these notes][4] (especially sect. 9) will be useful. 


  [1]: http://library.msri.org/books/Book43/files/andrus.pdf
  [2]: http://www.ams.org/journals/tran/1972-163-00/S0002-9947-1972-0292875-1/S0002-9947-1972-0292875-1.pdf
  [3]: https://www.ams.org/journals/tran/1967-127-03/S0002-9947-1967-0210748-5/S0002-9947-1967-0210748-5.pdf
  [4]: http://user.math.uzh.ch/stufler/lec/2018hopf/hopf.pdf