We get a lot of questions about member of the citrus family. Some varieties will flower and set fruit indoors. So, yes, you can grow citrus indoors but they are challenging to keep looking nice at the size of your plant without a greenhouse.
- is the pot the right size? This depends on whether it is pot bound or not. It sounds about right. This is part of the challenge of potted citrus. If the soil becomes alkaline due to the organic matter being used by the tree and the water you add then manganese and iron become unavailable in the soil and you will see nutrient deficiencies. The best way to tell is to see how firmly rooted the plant is in the pot. If it is firmly rooted then take it out for an inspection. A pot bound plant can be root trimmed have some soil added and go back in the same pot with an hours work.
- any time you have parts of a citrus plant that are not receiving high light then the foliage will thin out over time. The branches above and below the high light area will shed leaves faster and grow slower. You can counter act this by thinning the high light areas to let more light below and pruning the tree a little smaller.
With indoor citrus it is always prudent to keep watching for common pests. Spider mites thrive on citrus in the typical low humidity levels indoors.
The most common pest is scale of which there are multiple varieties that are controlled with soap and water and repeated applications.
You do not mention if this plant is from a grower or a seedling. Commercial citrus do not come true from seed. If your plant is a named variety then you could get some fruit from it. If not then it's hard to say.