Is there a premade solderable HAT-like board
Yes, tons of these are on sale. A quick search using words like "prototype" and "DIY" revealed there's a board called "ModMyPi", and there are certainly others:
There is a lot of Raspberry Pi 3 B cases but you want a case in which you mount Pi-Fan to that. In this case, open up this link:
TOPmountain Raspberry Pi Case Abs Protective Shell Case for Raspberry Pi 3 with Fan Hole(Black)
This case has a fan hole. To configure the Raspberry Pi for automatically turning on the Fan, follow this link:
I find that strip-board or perf-board will fill the gap between bread-board and a PCB (although PCBs can be relatively cheap using pooled services).
Some stripboards will have a special area on one end where you can attach a suitable dual row GPIO header (note that if you're not stacking them then you may not need a full size header, only one big enough to ...
First, if by "transportable" you mean you can put it in a car and take it somewhere, great, but if you mean portable like a phone, you are unlikely to get everything you want with it.
I would like to know how to manage the pinout of the Raspberry, since multiple sensors with their respective jumper cables are needed. Are there enough pinouts or can I add ...
The DHT22 doesn't use a specific protocol, so it can be connected to any GPIO pin not used by another sensor. Assuming your second diagram showing the 26-pin header shows the pins used by your RF sensor as those in lighter green with a label, any of the darker green pins without a label should therefore be fine for the DHT22.
As pointed out in a comment, +3....
from gpiozero import LED, Button
from signal import sleep
Then you can do
led = LED(24)
button = Button(23)
button.when_pressed = led.on
button.when_released = led.off
This makes things simpler, and it might just work.
And are you sure your wires are lined up with your button? In the real photo it's hard to tell. Maybe you ...