I would like to build a mini Ahrefs, where I can get some basic stuff (like Domain Authority, Page Rank and # of Backlinks) to any link I want.
Ideally, I would think, the simplest way would be to just ping the Google Search API and get that info. But, it seems the Google Web Search API is deprecated, and Custom Search seems to be focused on allowing me to search within 1 site or a small handful of sites. Plus the rate limit of 100 queries per day, is kinda ridiculous.
What's an alternative way for me to go about getting access to that data that I want?
Pagerank is no longer available to the public, Google has now switched off the API.
Domain Authority is not a Google Metric - service like Moz.com and Ahrefs.com construct their own metrics to try and 'weight' websites. You'll only get access via their APIs.
Number of backlinks you can try to get from Search Console. This is only accessible via their 'download latest links' csvs however. There is no API access beyond querying your search analytics.