I'm starting a Mac OS X app that will connect to a small website I'm currently developing.
The idea is that the app will store a number of snippets on a local database and then sync with the website whenever it can.
So my questions are:
Carbon or Cocoa?
SQLite for local storage / another solution?
Is there any decent free text-editor control with basic syntax highlighting to embed on the app?
At this point in each product's lifecycle, there is not really any compelling reason to use Carbon.
It's a fine API, and fine applications are written using it (Photoshop, iTunes), but a 64 bit version has not been made available, and it just seems that the writing is on the wall.
If you are just learning, and you want your knowledge to transfer to new projects in the future, Cocoa all the way.
Carbon is being phased out. You should really go for Cocoa.
Cocoa has its own database solutions (a flavor of SQL I think).
For new projects you should use Cocoa. Carbon is only useful to port already existing apps.
Cocoa brings it's own database backend called Core Data. It's quite simple to use but at the same time really powerful.
For syntax highlighting use a NSTextField, they are really powerful in Cocoa. One can even write highly graphical applications like Pages with it. For some examples of syntax highlighting see syntax highlighting on cocoadev.
You could also have a look at how other editors do it. For example there's Smultron, which is discontinued for the moment.
If you know ruby you can try out MacRuby. This let's you use ruby with the Cocoa classes. I'm quite sure that ruby will be the second programming language for the Mac because of its similarities to Objective-C.