To add to this, some ISPs ([1]used to) charge monthly fees for using their router or modem. A $7/month adds up, and in a year or two, investing in your own hardware will pay for itself and give you more control.
[1] 10 years ago, you had to rent the routers, but nowadays, the router could be free. You can bet though they’re getting that $7/month another way, and it’s likely because they’re selling your data.
I don’t think so, since the pi-hole (running on a raspberry pi or other computer) just acts as a DNS server which you configure as the DNS to be assigned to clients as they connect to the router.
If you’re not able to configure a VPN on the router, then setting up a VPN on the computer you’re using, not the raspberry pi, is the only option. This would only mask your computer’s IP address. This would need to be repeated on your phone and other devices as well.
A pi-hole simply black holes DNS lookups from known advertising networks and malicious domains, so your computer is unable to connect to those domains. This is good because you probably want to block those domains, but it doesn’t protect against everything. Adblocking in browser using uBlock Origin will achieve similar results, but only applies to the browser, not other applications on your computer, or say your phone or IoT device on the same network, which does DNS lookups via pi-hole. Both pi-hole and uBlock Origin do not provide any protection from hiding your real IP or your location. This is where a VPN comes in.
Personally, at the router, I black hole a minimal set of hosts from lists I know I will never want anything connecting to. For example, you could use one of the OISD lists: https://oisd.nl/. Then in your browser, you can add uBlock Origin and add more lists which you can selectively allow on websites. uBlock Origin has lists which block against internet annoyances, which pi-hole can’t block against (since it’s blocking DOM objects, and not DNS lookups). This is also useful because it’s easier to control uBlock Origin in the browser, and you can disable it for only some sites. Adding a VPN in addition to this satisfies IP and location hiding, which you can add on the whole router if it supports that, or just your computer/browser if you want.
<Incredibles Meme> Off means Off </meme>