Yes. At the moment, this is done using Aspire Snapshots, so both full crawls and incremental crawls will traverse the site or list depending on the root url, with the difference that during incremental crawls only adds, updates or deletes will be indexed.
For a discussion on crawling, see here.
Those permissions ("Full Read") are required so the connector can fetch additional information from site collections. This information is required for incremental indexing.
If setting permissions at Web Application level is not suitable to your environment, consider setting site collection administrator rights to the crawler account on each site collection you want to crawl. (This will work without Web App permissions, but has to be set on every site collection.)