Have Nuclear Arms Effectively Ended Major Wars?
Well we can all agree that the answer here is no as we have had war since nuclear arms have been invented and diseminated.

And would universal adoption end small ones?
No universal adoption would not end small wars.

The reason is that the use or lack of use of nuclear weapons is down to a carefull balancing by either power on it's ability to win or prevent defeat. If everyone has nukes then it depends on who you are allied with do you and all you allies have more nukes than you enemies. If you do not have more nukes than you enemy then you need to be sure you enemies will not attack, if you cant be sure then you may need to attack in order to show you would attack your enemies allies if they respond.

I effect you need to make you enemy understand you will respond, if they do not understand then the likelyhood of miscalculation increases.

Also the less nuclear weapons involved the more likely there use, if we all have only have 10 nukes then even a small countries might think they can gain advantage from there use.

The calculation works like this

The more nukes you have the greater the chance of use.

The less nukes you have the greater the chance of use.

Did those fateful August 6 and 9, 1945 bombings bring an end to the cycle of major powers directly engaging each other in conflict that had lasted since the dawn of humanity? Could the current major powers cut virtually all 'hard' military spending without jeopardizing their respective national securities? And by 'hard', I mean the legacy institutions meant to engage other major powers such as the standard infantry and armored divisions, the naval battlegroups, and the air combat commands, not the special forces anti-terrorism/anti-piracy small scale stuff.
Possibly but I doubt it you still need to hold the ground even after you fire nukes that requires armies.


Further, if every nation maintained a nuclear arsenal, would there be no more wars?
No civil wars can and still do escalate how do you use a nuke on you own country, you could but how useful would it be.