The most important safeguard against disaster is to make a backup copy of your entire FP2003 site before you begin, and never delete it. That way, you can freely try out the Dreamweaver changeover. If anything goes wrong, start over with a fresh copy.
The little globes on your FrontPage folders are an indication that Frontpage has configured them in Windows as "System" folders. In Windows Explorer, if you click View > Choose Details... > and enable the Attributes column, you'll see an "S" in the column.
That classification is a nuisance sometimes (a
dir command in Command Prompt by default doesn't show the existence of System folders, and to enter it, you have to know it's there or use
dir /as, and if I recall correctly, sometimes search programs aren't allowed to enter System folders to search their files).
If the System classification seems to cause any problems in Dreamweaver, you should be able to remove it using the DOS attrib command, something like
ATTRIB -S C:\[full path to your website folder]\foldername\
For help, in Command Prompt:
ATTRIB /?
or
HELP ATTRIB
I've saved the most difficult question for last, as I have no experience at all with FP subsites. After some research, the best sense I can make of it is this:
An FP subsite is a way of managing what other people call subdomains, from within FrontPage. It allows you to access
hxxp://example.com/blog/
as
hxxp://blog.example.com/
When using the FrontPage Extensions (FPE) on the server, they take care of the mapping of incoming requests to the correct subfolder of your website, because you've already configured it in FP.
It appears that the more common, non-FP, way of managing a subdomain is to just publish it as a folder of your main site, and then configure the mapping of example.com/blog/ to blog.example.com in your webhosting control panel.
It looks like you have two options for dealing with your current subsites. If it were me, I'd probably try them both, knowing that I could revert to my backup copies and start over if anything goes wrong.
Option 1) Don't do anything special in FrontPage prior to importing the site into Dreamweaver.
I suspect Dreamweaver will not know or be able to detect anything about the status of the subfolders as subsites, and all links between your subsites, or between a subsite and the main site may be considered broken because blog.example.com is unknown. That probably can't be resolved until the site(s) are live on the server and the subdomain configuration is fully set up. Even then, Dreamweaver itself will probably always consider the links broken (or at least not recognize them as local links).
I don't know what other problems you might encounter.
Option 2)
"Convert your subsites to folders" in FrontPage prior to importing the site into Dreamweaver.
FrontPage will give you some warnings about the consequences of doing this. Consider them carefully because they should give important clues about what problems you might encounter. Then go ahead and do it anyway because this is, after all, an experiment to see what problems arise.
I suspect FrontPage will, as it often does, globally adjust links (and the interlinking, if any, between subsites) so that the sites refer to each other as just pages in separate folders of the same website, not as separate subdomains. It might make other adjustments, too, which hopefully it will have warned you about in advance.
Each subsite will probably lose its theme and all other FP characteristics that it had that were different from the main site. That is, it's being merged into the main site.
The goals of these experiments would be to discover
a) Which method does not corrupt the site/subsites (or at least corrupts them the least) and makes them easiest to work with in Dreamweaver.
b) Which method works properly on the live server after subdomains are properly configured.
c) Choose the best.
Hopefully, one of the methods will work so well that it is obviously the right one to use.
With either method, if your subsites have FP password protection (enforced by the FPE), they'll lose it, and you'll need to apply new password protection at the server level, such as in cPanel > Password Protect Folders.
As mentioned, I've not done this type of conversion myself, but I hope that is at least accurate enough to help foresee some of the potential issues.