Basic text selection
According to the Zathura Wikipedia page:
Zathura can search for text and copy text to the primary X selection
This implies the ability to select text as you read is built in, though it likely requires your mouse (you'll be hard-pressed to find a solution for keyboard-controlled selection).
How minimalist do you need? I use Atril, a slightly lighter-weight fork of Evince (the GNOME document viewer). Atril was made as part of the MATE Desktop (a continuation of GNOME 2). It's pretty light, though it does still have a GTK+ dependency.
Another option is Xpdf application. See also Wikipedia's List of PDF Software § Linux and Unix.
Regex
The only (usable) regex search implementation I know of, aside form command-line tools like pdfgrep, is actually your web browser. This isn't so usable, but here's a solution in Firefox: Open a PDF in Firefox and open your Developer Tools Javascript Console (F12 or Ctrl+Shift+K). Run these commands:
» pdf = document.getElementById("viewer").innerText.replace(/[ \t]+/g, " ");
» function grep(what, context=100) { return pdf.match(RegExp(`[\\s\\S]{0,${context}}${what}[\\s\\S]{0,${context}}`), "img"); }
» grep("put your regex here")
» grep("get more context", 300)
Note that you'll have to escape your backslashes. The grep command has an optional second argument, the number of characters of context to provide on each side (default=100).
Chrome and other browsers with built-in PDF viewers should be rather similar, but you'll have to figure out what HTML object holds the actual PDF content (it's the id="viewer" element for Firefox, not sure about the others—in the worst case, just use document.body instead of document.getElementById("viewer"). You may match items in the table of contents.)