Is it possible to grow a tiny tree in a flower pot that can produce fruits in house environment? Specifically, I'm interested in tangerines, pomegranates, bananas and lemons. How do I turn their fruits into growing plants? Even if some or all of them can't produce fruits in a flower pot, I'm still interested.
I grew potted citrus trees in Connecticut and it was very easy. I grew regular varieties, not dwarf or semi-dwarf. I kept up-sizing the pots on a yearly basis, but with yearly root pruning, you could keep them smaller. I also moved them outside in the summer months, which they appreciated since they really like sunshine. Some varieties, like key limes are smaller in stature.
Also, I grew ornamental bananas in pots. Banana plants are less well-suited to container culture in my opinion. Two problems come to mind. One, a non-dwarf variety will quickly outgrow the pot due to the baby "sucker" plants that come up around the base and aggressive root system. One could prune those off, but unlike a tree, a banana plant is somewhat unwieldy to be trying root pruning that frequently. Secondly, due to the thin leaf structure they seem to be somewhat susceptible to spider-mites in the dryer winter months.
There are some dwarf varieties that can be grown in pots. One I've been looking into is a Meyer lemon. If you buy it old enough it will be able to fruit. I've read of people keeping it indoors all year round but most seem to bring it out during the summer. If indoors. It still needs adequate sunlight. I'm still looking into how much light is adequate. If you look around you should be able to find some other dwarf fruiting trees that can be grown in containers.
Yes you will need a dwarf variety or dwarf rooting stock to keep them small and "pot sized".
You might also want to look at similar but more obscure plants. Eg. kumquats instead of oranges.
Finally, think you will be out of luck with bananas. These are technically herbs and not trees, so dwarf rooting stocks aren't available. They are large plants.
I've successfully kept a calamondin tree indoors (which I bought when it was already at a fruiting stage). They look wonderful and the fruit tastes pretty good too.