How important are these for SEO purposes (discounting accessibility)? 1) Title tags on links and 2) ALT tags on image?
I personally never use title tags on links. It may be useful to add weight to additional keywords though. I think image alt tags are more important. They tell the search engine what the image is about. This is especially important when that image is also a link. Plus image alt tags can help get image search traffic.
Image alt is required for Image Search. Title tags don't have much value like before so it isn't much important for SEO but it is good feature to make your website lively.
We use them all the time because Google said to use them. Now, does it make a difference? No. But then, on inbound links, Yes. Therefore, IBL yes, internal links, no. But does Google give relivance to accessibility (including title) yes!