So, why does my local IP-address (IPv4) change always when I shut the router down and open it again?
I know, I suck at these kind of things 🙁
Don't be so hard on yourself. People *can* suck at things, but being new to something, or just not knowing it yet, that's different.
Now, about your question. It sounds like your router is the one issuing IP Addresses, so I'll go with that assumption. When your router issues an IP address, it basically issues four settings to your PC. An IP address, DNS server settings, default gateway address, and something called Time To Live (TTL). TTL basically tells your PC how long the address is good for. It can be days, weeks, months, or even minutes. If it's short, that might explain the new address every time. I think that's the most likely scenario at this time.
Remember, though, that the address given to your PC isn't the one Web sites like ours or services like YouTube see. That's a public IP address, probably assigned to your router, and controlled by the ISP. That (usually) is the one that Internet services and games are looking at.