Hi there, we've been working to furnish our new place here recently after moving back to town so I have just been going through this personally.
The recommendation of the second hand furniture market near Huangtupo is a good one - they have lots of stuff there, including non-furniture and non second-hand items (bikes, bins, appliances, etc.). Personally I find most new furniture stores are ultra expensive. Taobao is good if you know what you want and don't need to see it in person.
There is a huge furniture store complex (about 5 levels of 30 stores per level) in the east of town near the expo garden called 得胜家居.世博店 ... just say shízhálìjiāoqiáo (石闸立交桥 ... ie. "stone sluice-gate overpass") to get there, it's the whole eastern corner of the intersection (perhaps not immediately visible behind some trees). They have everything.
There are a bunch more around town but they are pretty equivalent.
You said you had issues with delivery costs quoted. If I'm not mistaken, any decent furniture store should deliver either free or very cheaply. They may however want to charge you by the truck rather than by the object, which is likely just a sales tactic. As a general upper bound on cost or failing that if you buy lots of stuff from different stores in the same area you can use a removalist company like 蚂蚁搬家 (mayibanjia: 'ant removals') who will do a quick trip with a few objects for about 400RMB or so, give or take, depending on hassle factor (access) and items.
Another option you can explore if you need quite a few pieces is to have someone build custom furniture for you. It's actually a lot cheaper than you might think (way cheaper than buying individual mass-produced factory type plywood crappo items in furniture stores) and you can get exactly the size you want.
Expensive to fly to Vietnam or Laos, Cambodia prices vary. Chiang Mai return including tax is 1700RMB at the moment (China Eastern). Bangkok is usually a bit over 100USD each way with Air Asia. Vietnam is cheap and easy to reach overland but it takes effectively 2 days or a full day and night to get to Hanoi. There is a new super fast train line from Yuxi to Hekou, not sure if it's open yet.
We had previously had experience and information regarding some hospitals in Bangkok as we lived there for a year until about a year before the birth.
Based on that first and second hand information we settled on BNH Hospital, which we felt maintained a comfortable balance between serving wealthier locals and foreigners and a correspondingly international standard of service. The location was relatively accessible too, we stayed down the road near Wat Khek which has good Chinese, Indian, Middle Eastern and western food. We actually did a bulk time deal with a 5 star place there (on the logic that a good breakfast buffet would be beneficial in late pregnancy) but moved in to a less over the top place shortly after the birth, then after considering and writing off the islands on account of lack of decent medical service availability extracted ourselves to Chiang Mai where there is less air pollution (for part of the year, at least) to await passports for our daughter that would enable us to return to China.
None of us had or have any medical insurance at all, and we did not find this was a problem, in fact it saved time and hassle we would otherwise have been lumbered with.
Our previous experiences included Samathivej and Bangkok Hospital. We decided against the Bumrun'grad hospital as both locals and foreigners asserted it was too comfortably ensconced as the 'go to' hospital for expats and inbound medical tourists (many from the Middle East), thus overpriced and relatively lacking in service.
We also flew my mother in law down awhile before the birth which was a memorable experience for her (lots of new food!) and a good way to deal with concerns from extended family.