I like your comment about how “rainy seasons” are almost always overstated. I totally agree based on my own experiences and research, although there are a few places where they should be taken more seriously than others. On my most recent December trip to Bali (2.5 years ago, I think), I spent 30 days there and the rain only became a problem that one time I mentioned. It was just an insane cloudburst while taking a minibus from Kuta to Ubud, and in the end it’s actually probably my best Bali story. Aside from that, it’s very humid during Bali’s wet season, but even that really didn’t slow me down. The thing is, in Bali you are usually on or near the beach or a hotel pool (feeling the breeze) or in your air-conditioned room. Or you might be touring around to see temples and rice terraces and whatnot, and everything is outdoors and beautiful. When it does rain it’s usually over in 15 minutes, but often the rain comes over night. That kind of humidity can be a killer when visiting a city, but for me it’s a small issue on a tropical island like Bali.
If you book a resort hotel from home you'll probably think Goa hotel prices are much higher than you were expecting, but if you book a family-run place once you arrive you'll still find fantastic bargains. It's probably wise to book a few nights online before you get to India, because it can be incredibly confusing at first. After that you might be able to find a similar place for half the price by just checking around the same area.
One reason that Vietnam isn’t as popular as it could be is that travelers often feel a bit hounded when they are there. For some reason, those in the travel business in Vietnam (working at travel agencies where you book tours and such) make tourists feel like they are being hustled. They are rarely 100% honest with what they tell you or about the prices they are charging. Still, it’s an amazing country and it’s also incredibly cheap, even if you end up paying a bit more than you were supposed to. I highly recommend it, and it won’t be a problem if you realize before you get there that you can’t take everything at face value in the way that you usually can in Thailand or Bali. Quality of hotels for the money is also very good.
For traditional holiday atmosphere, think about Rome or Vienna, with Christmas markets, nativity scenes, and decorated cathedrals. In Edinburgh, they know how to welcome the new year with several days of celebrations culminating in New Year’s Eve. For travelers seeking a wintery destination, Tromso in Norway has skiing, reindeer, and a chance to see the Northern Lights. Bundle up for a dogsled tour or join a whale watching excursion.
Also, as far as I hear, the worst that ever happens is they take your stuff. So if you leave your passport and most of your money in your hotel room, and keep your phone or camera hidden most of the time, there isn’t much to lose. I’m making it sound worse than it is really. I wouldn’t hesitate to go to Rio again, but of course I’d be aware of those few situations to avoid.
Southeast Asia is easier to deal with, and there are cheap flights connecting in the Middle East and landing in Bangkok, Kuala Lumpur, and Singapore. Each of those is a very interesting city, and there are many smaller places to go in the same region. In Malaysia there is Penang and Melaka. In Thailand there is Chiang Mai and quite a few others in that area. In Laos you can visit Luang Prabang, which sounds like it could be a winner for you. And there is also Siem Reap, which is home to the amazing Angkor Wat temple complex.