Instead of a GG ground sheet, you might consider going to your local hardware store and buying a little box of window insulation "shrink to fit" plastic. I think Frost King is one brand, I got my local Ace hardware house brand. There was a thread about this on this site you can search for if you like, but basically it's the same polycro. The benefits are (a) no shipping charge (but maybe sales tax) if you buy locally, and more importantly (b) you can get a larger size and cut it to fit. I did that with my Gatewood Cape; the size that GG sells isn't bad, but I find it handy to have a piece sized to fit the entire inside of the cape (which is indeed big enough to put my stuff inside). Mark one or two points to make it quick and easy to get it properly oriented inside.
In terms of ice axe vs. crampons (or both or neither) --- note that people's recommendations on this might vary some based on the particular snow conditions that they encountered. I went through in a relatively low snow year. Still lots of snow on the passes though, but my personal experience was that I needed neither axe nor crampons, and so mailed both home along the way. That Camp axe you're looking at seems like a fine choice to me, assuming that you know how to self-arrest and to make the assessment of when it's best to carry the axe (perhaps putting away poles to do so). I too personally tend to stick with poles --- I guess you could consider a self-arrest pole (no experience).
Bottom line is that if it looks like a lot of snow I'd still probably bring both a light axe and 6-point crampons, or maybe Kahtoola microspikes. Your resupply plan impacts this too a bit --- I resupplied in Independence, which gave me the option to mail some stuff home relatively early on after I had first climbed Whitney and gone over Forester Pass --- I.e., had a better in-person sense for what conditions were like.