I don't have good cell reception in my house and I'd to have my Google Voice number forward to my phone on Wi-Fi when I'm home. Can I just use Google Voice or do I need a third-party app to do the connection?
You need a third-party app to recieve Wi-Fi calls.
Two popular apps are GrooVeIP (paid & free) and Talkatone (paid). You could also set up Sipdroid (free), which will set up the SIP forwarding using pbxes.com. Or you could use a number of other SIP apps and set up forwarding yourself, but that gets a bit convoluted.
Yes, you can.
But the only GV function on Wi-Fi alone without some SIP server help will be texting and getting voicemail just like you can also get email over Wi-Fi.
I tried what you're doing last year just to get it to work. Using a free pbx service over the internet and GV, I was able to make calls over Wi-Fi. You need a SIP server or some VOIP server to 'find' your number over IP communication.
You could use Skype to do that also.
third party call apps for receiving calls via google voice have now been disabled. maybe GV will make an update to this. but with buzz about them phasing voice out.. who knows.