How to treat $\epsilon$ and '$' in top-down predictive parsing (predict table)?
$\epsilon$ is a terminal symbol


No.  $\epsilon$ is the empty string, i.e. no symbols at all.


  However, I've somehow interpreted that I could/should add '$' as a terminal symbol as well. 


Yes, $ is the special symbol that marks the end of input, and you may have to do things at that place and when you want to go past it, you are done (or in situation of error for LR parsing if the stack is not empty).


  The rules given here also treat '$' and ϵ as if they were separate:
  http://www.jambe.co.nz/UNI/FirstAndFollowSets.html


Considering $\epsilon$ as a symbol to be put in $FIRST$ could be an worthwhile implementation trick instead of computing $NULLABLE$ separately (I haven't tried it, at first sight it seems to break useful properties and to add as much complexity as it removes, see the point 4 in the computation of $FIRST$), but it is nothing more than an implementation trick.