Weakly normalizing + confluent = strongly normalizing?
Is it the case that a confluent weakly normalizing system is strongly normalizing?  Why/why not?


No, it's not necessarily the case. 


Weak normalization means: there exists a reduction strategy that will lead to a normal form. 
People often use confluence to mean what rewriting people call "local confluence": this is the property that if $a \leadsto b$ and $a \leadsto c$, then $b$ and $c$ can be joined -- i.e., there exists a $d$ such that $b \leadsto^\ast d$ and $c \leadsto^\ast d$. 


This does not rule out the existence of infinite reduction sequences. 

As an example, start with the STLC and add a new base type $X$ and constant $c,d : X$ with the reduction rule $c \leadsto c$ and $c \leadsto d$.

There is a weak normalization strategy: use a weak normalization strategy for the STLC and immediately reduce any $c$ to $d$. The calculus is also Church-Rosser, since it inherits this property from the STLC. But it isn't strongly normalizing since you can reduce a $c$ forever.