IIRC, the majority of Republican forces at the time were militia and volunteer brigades. Although I suspect some were formal army units, and some of the army units under Franco may have switched to them, I don't seem them having a serious standing army in the same sense the Germans did.
I think you are mistaken. More than half of the land army remained loyal to the republic, and almost the whole navy and air forces. The problem was that the most leftist elements of the Republic government were not quick to use them against the rebels because they were afraid they would switch sides in contact with the enemy (which was just an unwarranted, stupid fear).
Also, and with exceptions, the Republic decided not to start large-scale offensives, favoring a defensive war. Yes, exactly the opposite of what would have benefited them, as you have pointed out. Why? again, political considerations: the most leftist elements of the government wanted not simply to win against the rebels, but to turn that victory into a country-wide revolution. The moderate core of the government ended up agreeing, because they overestimated the effect of German and Italian help, underestimated their own forces, and by prolonging the war they were hoping to hook it up with the WWII, which everybody knew was coming.
Even if the Republican Spain had decided to be neutral, it's difficult to think that they could have remained neutral. Seeing how the government was had sympathies with the left, it would have probably been invaded on the same schedule as Barbarossa. An independent republican Spain was an obvious point for the Allies to start their assault on the continent. And even after the Nazis would have invaded Spain, it's more than likely that Avalanche would have taken place in southern Spain, rather than Italy.
I think the effect on the whole would be helpful to the allies:
-forces and resources spent by Germany securing Spain are not being used elswhere.
-a front in Spain by 1943 could be used as main western front, much better than Italy.
-the advance of the western allies could have been jumpstarted between 6 months and one year, making the disembark of Normandy a secondary operation to reinforce it.
-a faster advance of the Western Allies might have meant that the Europe under Soviet control would be significantly smaller.
For Spain it would have been a mixed bag:
-Being wiped by war 3 times over the space of 7 years: this is doubtlessly the worst, not only the dead in the wars, but also the lost generations, and the loss of harvest which could cause terrible famines all along the 40s (which were quite bad for agriculture due to persistent drought).
-Spain never leaving the international community, and receiving the American help from the Marshall plan, which could mean that Spain would have kept up with the rest of European nations.
Also, if the WWII in Europe finishes significantly ahead of OTL schedule... what would that mean for the Pacific theater? would have the US launched a land invasion before the nuclear bombs were ready?