Does the downloaded games stay with you after the plus expires? I cant afford to keep the subscription going and I was planning to buy it after i finished Yakuza 3 (Which I’m playing right this very moment, finding a copy was hell)

Once Playstation Plus expires you will no longer have access to play any of the games you got with the service. 

They will, however, remain tied to your account. So if you renew your PS+ in the future any games you had bought previously will be available again.