1. This can be controlled though file permissions. A file in unix has owner (the user who made it), group (a user can be in one or more groups, that can define extra rights), and other (everyone who does not fall into one of the above categories). If you own a file (say user a in this case made the file), you can right click on it in the Gnome GUI, go to the permissions tab, and uncheck the boxes by "Group" and "Other", so that only user can read the file, write to the file, or execute it (if it is a script or binary).
Note that the execute bit on folders gives the user who has that bit the rights to list the files in the folder.
2. There are a large number of groups, and they are required for services to operate, or to give you rights to use the services. For example, on my system I am a member of the cdrw group that gives me rights to burn cd's, the sound group so that I can play sound, and many others. If the groups were removed, then I would no longer have the rights.
Some services also require the groups, such as sqld is required for the sql server to run, etc. If they are removed, the services in question may stop working.
The way to look at groups is: if you did not make it, removing it may not be such a good idea.
Let me know if I am not clear on some points (I have been doing this for a while, so I may gloss over points that someone new to Linux would not know yet)