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.