Do Your Painting, Etc. Images Here Show Up On Google Search If You Do Not Offer Prints?

I don't use google - I wouldn't know. But when I do a search using BING - I get my IK images. I've not figure out how to change that - yet. Maybe it's why I've sold several there and only two here - about two years ago.

They should if you used keywords and wrote a description.

It shouldn't matter whether you offer prints or not unless you specified prints in your search query.

Yes they do. I just did a google search of you. It showed all where you posted your works.
etsy,artprice,linkedin,pet-portrait,etc.
Just be specific on what you want to search and it would show.
Whatever you post in the internet it would be there forever.