Which vpn are you using?

[Borges]

Quick question about VPN: let's say you use a shared wifi network e.g. from your employer or school. You have to register and provide your personal info to be able to use it, so the network admin can find out who you are. If you also use a VPN all the time you use this wifi network, how much can the admin find out about you from your browsing history?

VPNs are actually more useful for hiding traffic from your ISP or local network than for hiding your identity from authorities. VPN servers can be seized and you don't really know how much data they're logging - so they're a gamble when it comes to anonymity. However, VPN connections are encrypted and do effectively protect you from *local* espionage.

-Can they read and save your whatsapp or other instant messenger chat history? How about your personal email?

WhatsApp conversations are encrypted end-to-end. They can't be read by anyone but you and the recipient (assuming your phone isn't compromised). It doesn't matter whether you're using a VPN or not. Even the people behind WhatsApp are unable to read your messages.

Most e-mail services use HTTPS. HTTPS connections can only be read by your browser and the destination server. Again, the VPN doesn't really make a difference in here. That's what the lock in your address bar is for. If you're accessing an HTTPS website, the lock will show up. That means your connection to that website is encrypted.

-Can they find out if you visited a particular website such as this forum? If so, how much can they find out about you? Just the web address and the time you visited it? Your nickname as well? Keylog everything you wrote?

If you're using a VPN, they won't know you've accessed RVF nor anything else.
If you're not using a VPN, they will know you've accessed RVF and when. They won't know which individual URLs inside RVF you've accessed, nor who you are in RVF, assuming the connection to RVF is encrypted (HTTPS), which it should be.

-If you automatically back up your Android phone records, reminders, tasks, photos etc from the google environment, how much of this info is available to the wifi admin?

None of it, see the above regarding HTTPS.
In general you can trust companies like Google or Microsoft to do the right thing when it comes to encryption, data transfer, and storage. Your data can't be read by your network administrator.
Don't trust them to do the right thing when it comes to collaborating with authorities.