Hi there. These are some big questions! In fact, we are currently working on a book about the wheres, whys, and hows of this kind of travel - I will let you know when it is available!
I don’t mind talking cost, but the numbers for our whole trip won’t be reflective of yours - SE Asia was the cheapest part of our trip, by far. (We were in Vietnam and Cambodia.) I think Laos is also very cheap, with Thailand close behind. Singapore may be pricier, but I’m not sure, I’ve only been in the airport! I have no experience with Nepal, Malaysia, or India.
In Vietnam and Cambodia, you can live like a king for $50 a day…nice hotel with a pool, eating at nice places, having a few beers, taking taxis. On the flip side, I also know people who have traveled that area on less than $10 a day. Even 5 if you REALLY push it. All of this after the cost of getting there, of course. If you’re willing to do a shared room in a cheap hostel, eat street food with a lot of mystery meat, and figure out the public bus system, it can be done very cheaply.
When you’re thinking about cost, check into what immunizations are needed for different locations. We were surprised how much some of these cost, even if you have insurance! Visa costs are also an issue in some countries. Vietnam and Cambodia were $20 or $25 each, some countries are free, and others were up to $200!
We got interested in specific places by talking to other travelers who have been, but the details were all researched on line. http://www.tripadvisor.com/ and http://www.lonelyplanet.com/ are great places to read what other travelers are saying. We also used http://wikitravel.org/en/Main_Page quite a bit, and specifically answers a lot of the questions you ask.
For places to stay, I highly recommend checking out https://www.airbnb.com/. Much cheaper options than hotels, but a step up from some of the sketchier hostels. You can find really good deals if you are willing to take a shared or private room instead of a whole private apartment.
I hope this helps get you started. Stay tuned for the book, which will go into much more depth and detail.