To answer your questions:
Classical first-order logic, usually abbreviated FOL, has pride of place among the open-ended variety of logics that have been specified during the past century. Primary reason: FOL is sufficient to specify 99.99% of all versions of mathematics from ancient times to the present. FOL can be used to specify every digital computer ever built and every program that runs on any digital computer. And the syntax and semantics of any other versions of logic can be specified by mathematical theories expressed in FOL.
When I say that people in ancient times used FOL to specify mathematics, I mean that they used the equivalent of the words AND, OR, NOT, IF, SOME, EVERY, and EQUALS (=) in a way that could be translated to any modern notation for FOL, including eg1911. (http://jfsowa.com/peirce/eg1911.pdf )
Re Peirce's many versions of logic: Peirce made some extensions to Boolean logic in the 1860s, but his major extension beyond Boolean logic was his logic of 1870, which went beyond monadic predicates to n-adic predicates for any n>1. De Morgan called that work the greatest advance in logic since Aristotle. And he was right.
The discovery of complete notations for FOL by Frege (1879) and Peirce (1885) presented mathematicians with a logic that was sufficient to specify all of mathematics. That was a revolutionary advance. Peirce (1885) also specified a version of second-order logic. That was an important advance beyoind Frege (1879). (See http://jfsowa.com/peirce/putnam.htm )
Peirce also used logic as a metalanguage in his 1898 example of an existential graph that stated "That you are a good girl is much to be wished". These two additions (second-order logic and metalanguage) could be added to the eg1911 notation with the same or similar additions he used with the earlier versions of EGs.
The semantics of those additions could be specified along the same lines as modern extensions to the algebraic notations. One version I have been using is called Common Logic (CL). For references and discussion, see the slides I presented at a conference in June: http://jfsowa.com/talks/eswc.pdf
Re modal logic: Any of the notations for modal logic that Peirce introduced before 1911 could be added to the notation of eg1911. But Peirce himself was unsatisfied with them. He mentioned a replacement, which he called Delta graphs. But so far, nobody has found any MSS that specify any detail. But any extensions during the past century could be added to the notation of eg1911. For some discussion, see http://jfsowa.com/pubs/5qelogic.pdf .
Re three-valued logic: Peirce specified truth tables for three-valued logics in some MSS. Those could be used with the notation of eg1911. But the fact that he presented eg1911 at the beginning of a long letter on probabilty suggests that he may have been thinking of probabilty as the way to handle uncertain information. If so, classical FOL, as expressed in any notation including eg1911, could be used to reason about probabilities.
Unless and until any MSS after 1911 are discovered, nobody knows exactly how Peirce would have extended EGs to handle any of the above issues. But eg1911 is a *better* foundation for adding such extensions than any previous version:
1. The use of shading instead of cuts or scrolls supports a simple extension beyond a two dimensional sheet: just use shaded regions in N-dimensional space. In one of his MSS, Peirce explicitly said that selectives are necessary only for a 2-D sheet, and that EGs on a plane should be considered *projections* from 3-D graphs.
2. The drastic reduction in technical terms in eg1911 clears the way for further extensions. In L231, he mentioned "stereoscopic moving images" and regretted that he could not afford the technology. Today's virtual reality would be ideal for allowing anyone to wander through a moving 3-D graph and make dynamic changes to it.
3. With today's technology, it's also possible to include arbitrary images and even 3-d virtual reality inside any region of an EG. In a talk I presented at an APA conference in 2015 and later at an EG workshop in Bogota, I proposed two new rules of inference -- observation and imagination -- which could be added to multi-dimensional EGs. Those two rules would be special cases of the rules of iteration and deiteration. For the slides, see http://jfsowa.com/talks/ppe.pdf . Slide 2 of ppe.pdf includes the URL of a 78-page article that was published in the Journal of Applied Logics,