How to sort Google Scholar search results by number of citations

It had been gone in redesign few months before you asked this question and most probably you had enough time to assure yourself it won't return ever.

One mitigation would be adding those keywords from topic to search query text. The other could be limiting search down to particular publishing site with site:example.com/path stem, if you know that info in advance.

Google Scholar does not share DOI information anywhere, if they use it at all. There is currently no extension that does what you request. However, we can consider what it would take to make one.

We have three ways we might find the DOI from the information Google Scholar does provide:

1. We can start with the article title and search for the DOI using https://doi.crossref.org/simpleTextQuery or similar Crossref services.
2. We can load the article itself or its associated web page using the article link Google Scholar provides and look for the DOI somewhere in that document.
3. We can see if the DOI is embedded in the article link.

Humans can find DOIs for most articles using methods 1 and 2. A computer program can probably do the same, but this would require fetching a whole HTML document for every article we needed to look up.

For method 2, it is difficult for a program to distinguish the article's DOI from, say, the DOIs of cited articles, or other IDs associated with the article. Even with a good parser, there will likely always be some errors in this process. This many server requests is a too slow for search results. If accuracy is important, then our program could convert the DOI back into a citation to verify it, but this is yet another server request. However, such a tool might be useful to add DOIs to a bibliography.

Method 3 is very doable for a web browser extension, but I estimate that this will only work for about 40% of articles, such as those published by Wiley and Springer. Try, eg., $('a[href*="/10."]') on the Google Scholar results page in your browser Javascript console to see what you're working with.