Answers

Each case is unique but in general, the chimney should be done first. Special attention should be paid to the roof flashings around the chimney.  If the flashings are behind the stone then make sure they extend far enough into the shingles so the new roof can be flashed properly. If the flashings are surface mounted and there is a counter flashing then the roofer should have no problem making the chimney base watertight   

I am unfamiliar with this product but from your picture it looks similar to many of the screen-type products that are available on the market. As a roofing contractor who has been installing "gutter protection" for over 30 years, we tend to not encourage 'screen-type' products as the maintenance is difficult and frequent. We have been installing GutterTopper for over 20 years in Wisconsin and our customers are very satisfied with it's performance. All products require maintenance, but GutterTopper's is the quickest and SAFEST by far.

I believe a dealer in your area is Marshall Roofing.
http://marshallroofing.com/gutter-protection/

or contact http://guttertopper.com/ directly.

I hope this helps. The screen you are considering is much like ones we often remove because of customer dissatisfaction. 

Roof first, then foundation. (not saying that just because I'm a roofer :)
Gutters are likely involved too.

Fixing the foundation should not undo any efforts spent on the roof and gutters.

Was the product sold to you as "maintenance free"? If so, request service. If not, the maintenance of cleaning off the tops is your responsibility I believe. I would suggest not ignoring it. A few problems can arise, especially if you are in a snowy climate.