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At this size, you want a fertilizer that has an even NPK ratio and a low ratio. The numbers should 8-8-8 or lower. If it is a potted lemon, I would recommend using a water soluble fertilizer. This article should also help: https://www.gardeningknowhow.com/edible/fruits/lemons/learn-about-fertilizer-for-a-lemon-tree.htm

You should not harvest the first crop, but actually you should just remove the fruit as soon as you see it. When you first plant a fruit tree, it is best to let it focus on becoming established for the first year. When it grows fruit, its energy is split between fruit and root growth, rather than just focusing on root growth. Removing the fruit allows the tree to work on growing a big, healthy root system.

The amount of time time it takes for a lemon to ripen can vary depending on several things. First, different varieties have different ripening times (typically between 3-18 months). Second, the conditions the lemon is grown under can shorten or lengthen the time it takes for it to get ripe. In your case, I would suspect that the house is not warm enough to encourage ripening and it slowed down the ripening process when it was brought indoors. Once you set it back outside, the lemons should start ripening again.

It needs to be pollinated. Since it sounds like yours is in a container, it probably does not have access to pollinators. So you need to be the pollinator for it. It is very easy. Just take a small paintbrush and swirl it around the inside of the flowers. Don't pollinate all of the flowers, as the plant produces more flowers than it can handle if they all become fruit. Just do a few per branch.

These articles have information on how to fertilize your trees: https://www.gardeningknowhow.com/edible/fruits/citrus/fertilizing-citrus-trees-best-practices-for-citrus-fertilizing.htm, https://www.gardeningknowhow.com/edible/fruits/lemons/learn-about-fertilizer-for-a-lemon-tree.htm I would also try giving it some phosphorus rich fertilizer, like bone meal. If it is lacking phosphorus, it will not be able to produce blossoms (which means no fruit.). The requirements for lemons are very similar to that of limes. If you have not read it yet, this article may also be helpful: https://www.gardeningknowhow.com/edible/fruits/lime/reasons-and-fixes-for-lime-tree-not-producing-blossoms-or-fruit.htm

There is a possibility that they would cross pollinate but you would only see the results of that cross pollination if you planted and grew the seeds from the cross pollinated fruit. As citrus trees are not typically grown from seed, it is not normally considered an issue.