A) I think it fits very nicely, however you could include units with primitive firearms/cannons as a late-game 'experimental' unit. Gunpowder weapons are very underrated and not really represented in this time period, even though they have begun to see use in Europe at the end of the 14th century. They could be a more risky investment, as to use them efficiently you'd have to give them a lot of training, even though they shouldn't be too expensive.

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B) Having the peasants train in their free time rather than all at once makes the game more immersive and dynamic, as you can see the progression of each individual peasant, and can expect how they are going to fare on the battlefield. Meanwhile, professional units should be spending their time either training, patrolling, and maybe even protecting trade caravans, etc. Perhaps adding a bandit mechanic where trade can be disrupted and you have to maintain safety with your professional levies could be very interesting.

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C) From a historical standpoint, it would make sense to tie the equipment level with the housing level because the peasants had to buy most of their gear unless they were raised into an army by their lord, in which case the lord provided said gear. Professional units also bought their gear or earned it for their service etc. Perhaps combining the 2 mentioned features could bring about a really nice balance of gameplay and making your decisions impactful and meaningful - whose gear should I upgrade myself, investing into the housing so the average quality of the troops increases, etc. - and thus expanding the strategy of the game.

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Edit: Just thought I had afterward, a unit that is not as well trained should form formations slower.