Electronic – Why do heaters vary widely in resistance even though they have similar power requirements

This type of PCB connection is relatively easy to solder to - but it takes practice.

1) Small tip diameter. I personally use a Metcal sttc-126 tip, which is only about 0.016" diameter at its' smallest point. This particular tip runs at 700F and I normally use 63/37 tin/lead solder.

2) Tiny amount of solder. Although I regularly use thick solder (0.050"), I would recommend that you use solder that is somewhere between 0.015" - 0.031" diameter. Rosin flux works well. Avoid using no-clean flux - I find that it just isn't active enough.

3) Thin wire. Someone else mentioned 30 AWG wire, which is what I would use. Strip about 1/8" insulation from the wire, then trim the bare portion so that it is only about 1/16" of exposed conductor. Now tin the exposed conductor with solder.

Prep all the wires ahead of time. It makes the whole process quicker and you will find that attaching the wires to the PCB just 'seems' to go better as you go along.

Start by ensuring that your solder tip is clean and well-tinned. Support the PCB so that it is vertical, with the little wells of the connection at the top. Quickly, very quickly, use the soldering iron and solder and fill the little well about half full of solder. Each solder joint should take no more than 1 or 2 seconds. Do all of the connections on the PCB along that top edge.

Now take one of your prepared wires and lay the exposed conductor into one of the wells. Touch the soldering iron on top of the wire, with the side of the iron also touching the side of the well. If your soldering iron is well tinned, the solder in the well will melt within a second or two. Let the wire drop into the molten solder and remove the iron while holding the wire still.

Now do all of the connections, one at a time.

This skill takes practice. You mention that you have already wrecked a couple of boards. That's fine - use those boards to practice on.

Am I doing something horribly wrong

The op-amp chosen is wholly unsuited to your expectations. The gain bandwidth product is 2 MHz and that means, as a unity gain amplifier, it will have a bandwidth of 2 MHz. With a gain of ten is will have a bandwidth of 200 kHz. Withy a gain of 100 its BW will be 20 kHz. You expect a gain of 300!

Also, the amplifier step response settling time is 150 us typically.

If not, can I get faster output response from this setup

To operate the way you plan, you need a much, much faster op-amp or you need to find a way of clamping the heater voltage to virtually zero volts so that the op-amp isn't having to recover from saturation and deal with a massive step response.

Is there a more appropriate approach to doing this (in hardware perspective)

I would consider using a sample and hold technique so that the thermocouple is disconnected from the input just prior to the heater powered on and reconnected just afterward the heater is powered off. Use a 100 pF capacitor for "holding" the thermocouple voltage when not connected.