I am in Z5 and wonder if I can grow this as a house plant in the winter and have it outdoors when weather permits. Will it ever give me fruit?
Sure can and yes.
Yes. I have a couple of Meyers Lemons that fruit well here. In pots that get moved outside in the summer and back indoors as the weather cools. A few things you need to know about how to manage the location switch, but not a problem.
Before I built a greenhouse I kept my citrus in pots and moved them inside for the winter. I would rotate them from a sunny window where everyone could see them into a back room that was kept cool and with higher humidity. They didn't like the dry air of the house so they could only stay "on display" for about a month. They don't love being houseplants but you can do it and they will give you plenty of fruit.
From what I am learning, I can keep it outdoors in the spring/summer/fall and then indoors in a sunny basement window where it will be cool for the winter. I am assuming I don't need a pollinator for either a Meyer lemon or a lime tree.
You do not need a pollinator, which is great. But, once you have one citrus tree, you'll want another! Mrs. G