Can a Raspberry Pi boot with nothing attached except power?

You don't need a keyboard or screen at all. The Raspberry Pi Imager tool on your laptop handles the whole first boot now. You pick the OS, then click the settings gear (or hit Ctrl+Shift+X) and it lets you bake in your wifi name and password, enable SSH, set the username, even change the hostname. Flash the card, pop it in the Pi, power it up, and it joins your network on its own. Give it a couple of minutes on first boot since it expands the filesystem, then find it on your router's device list or just try 'raspberrypi.local' from your laptop and SSH in. That's the entire setup process these days. Years ago you'd have had to fiddle with manually editing files on the SD card, but that's all built into the imager now.

One thing to decide early: do you actually want wifi, or will it sit near your router where a network cable is an option? For a box that's going to live in a corner for years, I'd plug it in if you can. Wifi on the Pi 4 is fine, but a dropped connection on a headless box is annoying to debug remotely.

On cooling, for a couple of light tasks you'll be fine bare. The Pi 4 throttles itself at 80C rather than cooking, so worst case it slows down, it doesn't die. Mine ran a small web thing for a year with no heatsink in a drawer-like enclosure and never throttled. If you start seeing throttling you can check with the 'vcgencmd measure_temp' command over SSH, and a cheap stick-on heatsink ussually sorts it. A fan is only worth it if you're doing something heavy. For what to actually run on it, common first projects are pi-hole for network-wide ad blocking, a small file server, or Home Assistant if you've got smart home stuff. All set up entirely over SSH after that first flash.

This is basially what I did with an old Pi 3 last month, except mine gave me headaches untill I figured out the power supply was the problem. I had it on some random phone charger and it would just randomly reboot or act wierd for no reason. Swapped in the proper official supply and it's been solid since. So if you dig that Pi 4 out of the drawer, check what you're plugging it into, that part tripped me up more than the setup did.

One thing I learned running a Pi headless 24/7: SD cards are the weak point. Mine corrupted after about a year of constant writes from logging. Booted off a cheap USB stick drive instead and it's been fine since. The Pi 4 can boot straight from USB, you just enable it once over SSH. Worth doing if this thing is going to live in a corner unattended, since you can't exactly watch it die when there's no screen attached.