Filemaker Client will happily host your system (up to 9 other users.)
You cannot have the same FM client twice on the same network.
IWP is really good. There are a few scripting limitations (not major, and well-documented) which do not work 'natively' in IWP, but if you know your file will be accessed by IWP you can easily accommodate those when designing the scripts.
All the data will be available to all the users, no complex set-up required.
You don't need Advanced to accommodate sharing; it has developer tools that will be handy for you when designing. There is no difference from the users' viewpoint.
There is no rapid application development database to match Filemaker. (Well, what did you expect...? But then you haven't actually told us what the job is that you want it to do.)
One FMP serial number will allow one user to access and share the database.
A second copy with the same serial can not access the hosted database on the other computer.
Up to nine FMP users (unique serial numbers) can access the database shared using the built-in sharing in FMP and FMP Advanced.
You could have three users with the same serial number and any ONE at time could access the database.
FMP or FMP Advanced can host a database(s) via Instant Web Publishing with a limitation of 5 concurrent users.
FileMaker Server Advanced can host a database(s) through Instant Web Publishing for up to one hundred.
Google "filemaker iwp limitations"
Filemaker is the most popular cross platform easy database currently.
The advantage to having FileMaker Advanced is for you as the person responsible for creating and maintaining the database. The script debugger alone can save you hours when a script does not produce the expected results and you need to track down the reason. Throw in the database design report, data viewer, the ability to add custom menus and custom functions to your solution (and which others without FileMaker Advanced can use once you have added them to the file), and you have a tool that can save you time and produce a much more professional user interface.