DIY Power supply kit suggestion (Fan & pull-up resistor, cap)
R1
You name this a pull-up, but if it's really between a 7824's Pin 1 (input) and Pin 2 (ground), it's an input-loading pull-down resistor. Since you don't change the input voltage, don't change the resistor.
Replacing 7824 with 7812
That means you'll have 12 more volts that the resistor has to drop. Also, a 12 V fan of same power as the original 24V fan needs to draw about twice the current – in sum, you will have more than twice the heat produced in U4.
It's unlikely that the heatsink was sufficiently oversized (these things are relatively expensive) to support that.
So, don't do that, unless you want to toast your 7812. Instead, you'd connect a switch-mode voltage regulator specified for the input voltage, 12 V output and the current you need. It's easy when you have such through-hole components – you can simply solder in wires to a sufficiently beefy buck converter module (don't go ebay/aliexpress on that – only buy things with proper datasheets).
heat sink &quot;plate&quot;
That's an isolating barrier that still lets thermal energy pass, typically made from mica. Yes, you'll need that if you want to attach your heatsink to a 78xx-style regulator.
2200 µF instead of 3300 µF
Obviously, someone designed in a 3300 µF capacitor because they decided that using a smaller value wasn't appropriate. That alone should tell you that, no, just reducing capacitance by 50% isn't a good idea.
Then again, this is just a buffering capacitor to smooth us the irregular and potentially very noise current draw of the fan. It is kind of usual to slightly overdimension that to allow for degradation and variability and noise-sensitive environments, so you might be fine. However, again, when going from a 24 V fan to a 12 V fan with the same power, you're doubling currents and hence probably also significantly increase current ripple – and that might be so much that it makes the life of your regulator hard.
So, I'll go with a &quot;no, unless you know very well through measurements that the ripple you see is negligible and you don't run into EMI problems, you can't just replace a component with one 2/3 of the values&quot;.