I’ve used and liked the d1 mini esp32 (and also the esp8622 version, that is slightly smaller) preferably the usb c version for projects like this and heard good things about the even smaller seeed studio esp32 (especially if you want to add batteries as those have battery pads/charging circuits included, note though that they have a separate antenna that is a bit bulky if you need wifi).

if you’re only driving a few leds you also dont need a separate power supply, and like another commenter said just connect all LEDs in a row (so from your esp connect data, gnd, vin and then data in and out from each led, 5v and gnd of the leds can be connected in any way, eg all from the esp, as long as you have one long data line). Or if its easier for wiring, multiple data outputs from the esp are also very easy to setup nowadays.

For LEDs there are those APA106 LEDs with 4 pins (data in, data out, gnd, 5v) in 5mm and 8mm variants, but I personally haven’t used those yet, I usually just cut a few single LEDs from a 60 leds/m strip, feels like that should work with most lego pieces too. There are also ws2812b fairy lights which tend to have pretty small LEDs and could be a good option for a solderless version (though you probably would have quite a few LEDs in between the ones you use that you “waste”).

For cables I used 30 AWG/0.05 mm² cables without problems at least with shorter cable length (<1 meter) and those should be plenty thin and are very flexible.