LED Strip / Resistor Heater + Driver Circuit
Here, the transistor is biased through a potentiometer. 


There are a couple of problems here. 


The configuration you have chosen is called an emitter follower because the voltage at the emitter will follow the voltage at the base less about 0.6 to 0.7 V.
If the wiper is turned up to, say 10 V the emitter will try to rise to 9.4 V. With R2 = 100 &Omega; the current will be 94 mA shared out between all your LEDs.
Parallel connection of LEDs is generally a bad idea. The ones with the lowest Vf (forward voltage) will hog more current than the others and run hot decreasing the Vf further, etc. You won't get even lighting. Usually resistors are added to prevent this.
Series connection of the LEDs is more efficient as the same current can power more than one LED.



  I would like to have a linear (or almost linear) relationship between potentiometer value and brightness/heating (current through collector). In simulations, I can tell that for this circuit, the relationship is not linear. Is there a simple way to rectify this?


Yes.







simulate this circuit &ndash; Schematic created using CircuitLab

Figure 1. Current control.

How it works:


R1 applies a voltage between 0 and 4 V to the non-inverting input. Let's set it at 3 V.
OA1's output will rise until the feedback voltage on the inverting input equals that on the non-inverting input. This will happen when 3 V is dropped across R3 which in turn requires 100 mA through R3.
The current will be divided between the two strings of LEDs and R4 and R5 limit the difference in currents between the two strings of LEDs because if either chain starts to hog the current the voltage will fall on that string.



  For the dew heater, I require it to produce about 5W of power max. 


Your requirements aren't clear here. If you use a dimmer then you'll have heat dissipated in the dimmer and that may affect what you're trying to do. The simplest would be to have switches for the heater resistors.


  How can I calculate the values of every resistance in the diagram to ensure this? 


You should be able to figure it out from my explanation above.


  Also, since the resistors are in parallel, the sum of all currents will go through the transistor too.


Correct.


  Is there a generic part number for a transistor that would be better suited for this task instead of having a fried transistor? 


Look at the specifications for Darlington transistors. Work out at what value of current and voltage across the transistor that the power dissipation will be maximum and then figure out your heatsink requirements.


  Also, how much should the power supply be rated for this? I have an option to use either a 5V or a 12V power supply.


Decide on a voltage requirement based on the worst case of the LEDs or resistors. Figure out what current is required by both loads. That's it!


  For the flat field generator, the problems are even worse. Linearity issues aside, my panel will use LED strips like THIS one. I can't find a datasheet ...


"No datasheet? No sale!" or be prepared to spend a lot of time characterising them.


  ... for this LED strip but it has 18 SMDs per strip and runs on 4V. I think all 18 LEDs would be wired in parallel since white LEDs have a forward voltage of about 3.3V.


Correct, so it's not a good design.


  If that is the case, each strip would draw about 20mA x 18 = 360mA. 


You don't know that. There are no specifications.


  My panel will house up to 25 of these strips. If I connect them in parallel, they will draw 9A of power!


Amperes are a measure of current, not power.


  Is there a better way of doing this? If not, are there any generic transistor part numbers that will handle this amount of current, or will I have to break the circuit up into multiple drivers, each driving a few strips? Also, I doubt I will find a 12V power adapter rated 9A.


Your maths may be OK but you still have no specifications. 9 A won't be a wall-wart. It will be a proper power-supply.

Finally,


  ... about 0.3W of power, and spaced roughly 1cm apart.


We don't say "0.3 watts of power" in the same way we don't say "1 cm of distance" or "100 kph of speed". The watt is a measurement of power so we just say "0.3 W".