Do I need to use resistor between output shitft register and Darlington array?

No resistors needed.

The ULN2803 is not just a darlington array it has some input logic, see the data sheet.

The data sheet says:-

The UP/DOWN counter value (D) starts at 40H at system power ON.

So no it does not persist.

Sorry if this is too much of an insult but have you connected all the earths on the unit together. These are the points marked with that triangular type symbol and should all be connected to the 0v line. I say this because my students often didn't understand that was what the symbols meant.

It is like saying I have only got 19 pins on the arduino so I can only control 19 digital signals.
You can control as many SPI devices as you want, all you have to do is to sort out the enabling of the one device you want to talk to at any one time. This can be done inside the device, if it offers and address structure or externally by routing chip enables controlled from other I/O chips.

About 180 times per second is what I would like.

Yes this is still very slow for a processor.

What is the difference between P-Channel and N-Channel?

Google it but basically P-channels pull up and n-channels pull down.

What is a drive transistor?

A transistor that is used to drive a load. Drive:-  to power or to control.

Can shift registers work fast enough for the 24 transistors?

180 time a second for 24 outputs gives you 4320 pulses per second.  So this is 231uS, one instruction on the processor takes about 0.625 uS (yes I know about it doesn't always) so that gives you in the order of 300 instructions executed for each output. What else do you want to do that is taking time? Without this information we can't tell if you have enough but I suspect you do.

about 10x per second.

This is considered to be a very slow rate so yes what ever method will do.

I use logic level N-Channel mofset transistors for all 4 transistors in an H-Bridge?

Most designs use 2 N-channel and 2 P-channel FETs. But it wouldn't supprise me to see one with all N-Channel, although you would need a few driver transistors or FETs as well.

Lots of ways please look in the playground for information.http://www.arduino.cc/playground/Main/InterfacingWithHardware

If I use a serial connection through the two analogue ports

How are you going to do that then? Software serial?

will it be as fast as the I/O ports?

No because you are sending serial data and that takes time. In a 9600 baud system it only sends 9600 bits per second and that has to come out of an I/O port so how could this possibly be as fast?

Go on give us a clue, what is it doing that you don't like.

If you want to make the arduino work like a USB keyboard recognised by the PC as a USB keyboard device without any extra software on the PC side then the answer is you can't do that.

This is because this requires the PC to see a USB HID device. The arduino looks to the PC like a serial port not a keyboard. Therefore you need some software on the PC side that looks at the serial port taking data from the arduino and some how squirts this information into the PCs input stream.

Alternatively this device:-
http://www.create.ucsb.edu/~dano/CUI/#CUIv1/
is an arduino bolted onto a USB host, you might be able to do it with that.

Yes it is possible but it is a much more difficult project than you think. The RAM in an arduino is limited and so the time resolution / length of recording possible is not as much as you would like.

It's not as simple as that see:-
http://www.arduino.cc/cgi-bin/yabb2/YaBB.pl?num=1271712343

"What wire for heating element?"

No I disagree, wire is the wrong word thread is the correct word. Basically it is a fabric based conducting element that is needed not a metal based conducting element.

These can be based on polymers or carbon. In order to generate heat efficiently that have to have a higher resistance than the "wires" used to connect them to the battery. That link you posted for conducting thread has quite low resistance and while that would work if you put enough current down it, there would be as much heat generated in the wiring to the point you wanted heating as the bit you did want heating.

Sorry but I don't have a solution, however there was an instructables article recently on conductive flexible "ink" which was a mixture of carbon and "liquid tape". That might serve as a basis for what you want.

As was said you can't drive a matrix from two TLC5940s. However you can multiplex one of them to drive a matrix, like I did here:-
http://www.thebox.myzen.co.uk/Hardware/Mini_Monome.html
But if it is just a single colour matrix it is a bit of a waste using this chip.

For connections to arduino u should use a resistor... so that in case of a mistake u cant fry ur arduino or ur sensor... 10kR should suffice..

I would disagree, 10K is to much, you run the risk of not being able to pull down to a logic zero level. I would not use any series resistor to connect internal parts. Even if it doesn't stop it working it will severely reduce your noise margin.

wow! so i got some rare items "non-standard resistors"

Yes it is not often that I am surprised but now I am. In over 50 years of electronics I have NEVER seen a 50K resistor. (you didn't photo shop that did you  )

So sorry yes you apparently can have a 50K resistor. I will now show that photo around the lab.

P.S. We will be moving our manufacturing to Thailand later this year, you never know I might get a visit.