It is accepted that playing a D in a C chord is a "second", but a C ninth chord uses the flattened 7th to take the D note into the next octave, i.e. C-E-G-Bb-D is a ninth. This gives the chord a more altered (jazzy?) sound. You may omit any notes apart from the Bb, otherwise it's just a C second with a different voicing.
 
Tim, a flattened C13th would be played C-E-G-Bb-D-F-Ab. The D and the F can be, and often are, omitted, but the Bb must be played otherwise if you play C-E-G-Ab, it's just a flattened 6th and has its own sound. As soon as you add the Bb it takes the Ab to into the second octave and becomes a 13th. You can play these notes wherever, and in whatever position you choose, that's just voicing.
This is the progression,
C   D   E   F   G   A   Bb   B   C   D   E   F   G   A   Bb   B   C
1   2   3    4   5   6   b7   7   8   9        11      13        15
You can see that in order to create the higher numbered chords you must add the Bb in the first octave. In theory, you add the notes in major or minor thirds from the Bb, to get the higher numbers. It is also not unusual to omit notes you don't need. i.e. a C15 is C-E-G-Bb-D-F-A-B, but you leave out the D, F and/or A depending on the sound you prefer.
 
Good luck
 
How are you progressing Sharke, have you made any progress?
 
I honestly can't think of any shortcuts to this process, I still think it's easier to learn this stuff on a keyboard as you can visualise everything a bit easier because it's all in a straight line and you have the black and white notes as points of reference. 
I think you may have answered your own question when you said you learned the recorder early in life and it taught you to read, it's something you "learned" and will always be with you. Learning notes is just the same process.

gswitz
For memorizing anything, I tend to use this...
http://mnemosyne-proj.org/
 
It's an open source project that lets you build your own flashcards. Then as you go through your cards you click different numbers to indicate how well you knew the answer and that controls how quickly the card will cycle back.
 
You can organize the cards into groups and then activate only certain groups. So for example, you could group by tone count - 2 tones - 3 tones - etc. or by Key.
 
You can store all card as front to back - back to front or just front to back. In other words, you could have Amaj7 on the front and A, C#, E, Ab on the back. You could have it also show you these tones on the first, and you have to say Amaj7.
 
If you click through the cards every day, you'll be amazed that the information slowly begins to stick.
 
You can also export your cards to your phone and practice on the go.

That looks fantastic, what a shame it's not available for iOS though...it would be ideal to have something like that on the go. I did however find some flashcard apps in the app store which I'll be checking out.