Consequences of Polynomial & QuasiPolynomial time algorithms for UP, FewP and CoC=P I have came up with two alternate universe scenarios that I need some help figuring out the outcome of. Thank you in advance.

There exists two complexity classes $\mathsf{A}$ and $\mathsf{B}$. By definition $\mathsf{A}$ $\subseteq$ $\mathsf{B}$ but then it can be easily seen that $\mathsf{A}$ is truth table reducible to $\mathsf{B}$. Furthermore, $\mathsf{B}$ $\subseteq$ $\mathsf{coC_{=}P}$ (Zoo) and $\mathsf{FewP}$ $\subseteq$ $\mathsf{A}$ (Zoo).

1)What are the consequences of a Quasi Polynomial Time Algorithm for $\mathsf{B}$ ?
a)Does it put $\mathsf{A}$ in $\mathsf{QP}$ as well?
b)Other than proving that $\mathsf{P}$ $\neq$ $\mathsf{PSPACE}$ and $\mathsf{P}$ $\neq$ $\mathsf{PP}$ does it also show that $\mathsf{P}$ $\neq$ $\mathsf{NP}$?
2)What are the consequences of a Polynomial Time Algorithm for $\mathsf{A}$? 
a)Does it put $\mathsf{B}$ in $\mathsf{P}$ as well?
b)Since $\mathsf{P} \subseteq \mathsf{UP} \subseteq \mathsf{FewP} \subseteq \mathsf{A}$ we will have $\mathsf{P} = \mathsf{UP} = \mathsf{FewP} = \mathsf{A}$ . Then can it still be the case $\mathsf{P}$ does not equal $\mathsf{NP}$? 
c)Does it also show that $\mathsf{NP} \subseteq \mathsf{RP}$ since now  $\mathsf{RP}^\mathsf{UP} = \mathsf{RP}$?
1a) yes by the inclusion.
b) Not sure what you mean by "it". If P=PSPACE then A and B are both in P.

2a) Not necessarily, A could be much easier than B.
b) There are relativized worlds where P=FewP&lt;>NP.
c) No, RP^UP does not necessarily contain NP. Don't confuse UP with Promise-UP.