I've been working to fully understand it myself and I've concluded that too an extent the more I learn the less I know. But I'll do my best
Is it just a rebranding of the Tea Party or is it something completely different?
Likely only the most politically astute and fringe right of that group. I don't think the common tea partier would even know the term "alt-right" if not for Hillary. Tea Partiers are likely the MAGA hat wearers you mention below with no "alt-right" connection.
Did the alt-right start as a bunch of anime nazis and expand to more conventional conservatives?
I don't think most are Nazi's although there are undoubtedly some involved, just as there are Marxist/Lenninists on the extreme left.
Mostly, I think this is a group of millenials from 4chan (use of pepe is an indicator) that like to say all the wrong things and break all the rules in order to offend because they get a kick out it. It really is for their own entertainment. The amazing thing is how effective they were in attacking Hillary. Meme magic is real.
Was the alt-right just another name for MAGA hat wearers and the nazis are trying to coopt it?
No, I don't think so.
In a way I think the alt-right is largely a response to the social justice warrior movement and people wanting to look up the reality of what they see in daily life vs that which is being preached to them every day by the MSM. BLM, white guilt, white privilege, LGBTQ discrimination, male privilege, race and gender as a social construct, etc. etc. etc.
What they found were certain intellectuals that have been blacklisted for non-conformity presenting science, studies, facts, etc. showing that race and gender are a biological fact, men and women have different strengths and weaknesses as do races due to evolution (which most don't view as creating superiority for any one race), comparisons of the national crime victimization survey and prior conviction rates in the participation of crime and sentencing, etc. Any debate regarding the above aside, this will almost undoubtedly always result in extreme fringe movements like stormfronters, 1488er's etc. wanting on board.
After Hillary went on the attack regarding the "alt-right" it was the first time many people had heard of this "movement." This is where I think the "natural conservatives" came in to play in big numbers. Yes, they are mostly "rural," working class, low wage, whites that are sick and tired of the media telling them they are dumb, don't know what's best for them, instinctively racist, should be ashamed of being white, have no culture to be proud of because, well, as white's they don't have a culture, that they have white privilege even though most squeak by hand to mouth on a monthly basis, blah blah blah. I firmly believe many of these people probably even voted for Obama at least once if not twice.
All that rambling and creation of run-on sentences aside. I could be completely off base in all of this. In fact, I may change my mind by the end of the day due to some new revelation or piece of information being presented.
Oh, and many of the alternative media that are being labeled "alt-right" like Stefan Molyneux, Gavin McInnes, Mike Cernovich, Paul Joseph Watson, Milo Yianoppolus are not alt-right and have said they are not from the beginning.
You labeled me as such and it's not true btw. I'm not alt-right. At one point I thought perhaps I was. Not any more.