Alaska airlines/Virgin America Merger

When airlines merge, do the partner airlines come across also?

I'm an AS member and see that Virgin Airlines are partners with Hawaiian and Virgin Australia

I'm not sure the answer to your question but I found this article and it addresses some codesharing information:

http://www.usatoday.com/story/travel/flights/todayinthesky/2016/12/06/alaska-airlines-merger-virgin-america-gets-ok-doj/95049528/

This is actually the first I've come across this deal, as neither are big players in my region. But I just booked a flight with American and the first flight is operated by Alaska...in about six months. From the article, only codeshare flights that create competition between VA and AA would not be permitted from Alaska's POV.

It depends on the situation- there is no one hard and fast rule. In your case you seem to be mixing alliances and mergers. Each alliance and airline will be different. For example, Hawaiian had an "alliance" with UA. Not any more.

Plus, each airline may have different participation levels in the alliance. In the Star Alliance, ANZ and Singapore Air have somewhat limited participation, particularly when it comes to award flights.

Alaska has steadfastly declined to say what partners would remain, or what will happen to their frequent flyer program(s). The deal was only given the green light by the Department of Justice earlier today. News on timing, FFP, partners, which codeshares are to be eliminated, etc., will likely be forthcoming over the next few days/weeks.

When airlines merge, do the partner airlines come across also?

I'm an AS member and see that Virgin Airlines are partners with Hawaiian and Virgin Australia

It depends. Many airlines are part of a major airline alliance, like Sky Team or One World, but may also have partners with whom they share code share flights but are not in an alliance together. When two airlines merge and have different partners or are in different alliances, it's anyone's guess as to who gets kept as a partner. I mostly fly Delta and back when Continental was an airline they were Sky Team alliance partners with Delta. When Continental merged with UA, the new merged airline took the United name and the Continental logo and as far as alliance membership, they went with UA's Star Alliance rather than Continental's Sky Team. So generally speaking, no more Delta codeshares with CO.