If you use the latest BC plugins they hook into the .dwt files of Dreamweaver. BC though uses .html files for its templates. The BC plugin converts .dwt files during uploading into those .html files.
You wont be downloading .dwt files in BC at all, it does not suppor them, You will be reverse publishing them and DW again will be converting them back into .dwt from .html.
I do not use DW myself for many reasons, including the crazy .dwt file format for example but if you use other applications like I with Coda 2 you only use .html which is what BC uses for its templates.
Not sure why you have two different things there but sounds like you are viewing online and just downloading from BC rather then the correct method to have the dwt conversion DW and the BC plugin use.
Ah ha! As soon as you mentioned 'BC plugin' I realized what the problem is. My work computer (where I work on site 'A') can't connect to most Adobe services (I still haven't gotten any help with that...) including DW's BC plugin. I even had to set up the site I created on the BC website manually because DW won't log into BC.
I used remote desktop to check my home computer and log into site 'A' there and sure enough it sees my templates as .dwts just fine.
Here is the thread I started about the connection problems. If you happen to have any advice for getting that fixed I'd appreciate hearing it! http://forums.adobe.com/message/5167681#5167681
Dont use DW as I mentioned, so I cant help on that sorry.
On your work computer, you said you set up the site connection manually. Check your FTP settings, and at the end of your username, make sure you've added "/dw" to the end of your FTP username, like this:
If you have the "/dw", BC will send .dwt templates. If you don't have the "/dw", BC will send .html templates.
Hope this helps!