Lirghtroom 4 Catalog.lrcat is not a folder. (Or at least, it shouldn't be a folder) It is a file which contains your Lightroom database. Everything you do in Lightroom, all of the editing, all of the metadata, and everything else you create (but not the images themselves) is stored in the database. Thus, Lightroom (via the database) must know where your photos are at all times. If you move or rename a photo outside of Lightroom, then Lightroom cannot work with this photo (although you can fix this very simply). The recommendation for beginners is to do all photo management inside of Lightroom. This is a crucial understanding to your proper use of Lightroom.
Your .lrcat file(s) can be on any hard disk, external or internal. The usual recommendation is to put your catalog(s) on the fastest hard disk, which is the internal disk.
Some people (including me) think one large catalog is the way to go. There are many advantages. But other people think multiple catalogs is the way to go. In your case, in a photo is going to be used on multiple web sites, then I would recommend a single catalog. If a photo can be used on one (and only one) web site, then multiple catalogs probably works, one for each web site.
I'd prefer to export all my images from Aperture into Lightroom in a library (or libraries) in my internal hard drive - for now. When it gets too big, then I can move it to an external hard drive. Is this possible?
To me, this sounds like you are creating extra work for yourself. If you think that at some point you are going to want your photos on an external drive, just put them there to begin with. Then you don't have the extra effort of moving photos from here to there (but yes it is possible). In my opinion, there is almost no performance penalty for having photo on an external drive.
You might want to consider watching some introductory videos about Lightroom, for example
http://tv.adobe.com/watch/creative-suite-podcast-photographers/lightro om-dude-where-are-my-photos/
I can't answer questions about Aperture.