No. The 14/3 will either contain two travelers and a neutral or two travelers and a hot/switched hot. You will not have an always hot and a neutral available to run your fan. You will most likely need to come from one of the switch boxes to do what you want.

I would be interested in knowing if anyone has really done what mclarke is asking: changed a stationary pot light into a directional light; it isn't the same as changing the trim ring. I wanted to do it, and DH went in search of a replacement and could not make it happen, at least not simply - the position of the clips holding them in place were different.

First, regarding your recessed cans, what size cans are they? We installed 5" incandescent cans (Juno's). The rough rule of thumb is to place them somewhere between the edge of the counter and 3" from the edge of the counter. The cans themselves should be 4'-5' apart, unless you are getting the small 4" cans, which need to be closer together. As to your undercabinet lighting, whoa, stop right there! Why on earth would you spend money on undercabinet lighting, and not get good task lighting from them? More info please. What kind of undercabinet lighting are you getting? There are many choices available, many that give great light, without getting hot, and are dimmable, and at several different price points. We chose Juno xenon undercabinet light strips and are very happy with them, but there are other choices out there. If your cabinet guy (and, BTW, why aren't you talking to an electrician rather than your cabinet guy about this??) means that your undercabinet lighting won't be enough to light your kitchen, well, of course, that's correct. But if he's saying you shouldn't count on much light from them even for task lighting, I would definitely buy something else. While you're at it, make sure the undercabinet lighting is hung towards the front of the cabinets, not toward the back. If you install the lights toward the back, you are mostly just lighting the backplash, instead of shedding light on the counter. It's a waste of the light. For some reason, a lot of contractors insist on hanging them toward the back--it's an easier install that way, though not by much. So you stand firm that the undercabinet lights be hung more toward the front, end of discussion.

Modern Farmhouse is a broad description and it may mean different things to different people. I would look at modern farmhouse as a mixture of the clean lines and crisp geometries of contemporary architecture and a traditional farmhouse... The irregular pentagon is the shape that I most associate with the style. So when I think of interior ceiling features I lean towards features that carry the irregular pentagon inside. Such as this... In my opinion as you add beams to the ceiling you are removing the clean lines and crisp geometries that make it a "modern" farmhouse and moving it to a newly built farmhouse style. Both are quite common but if I wanted a mix of contemporary and farmhouse then I would opt out of muddying the visuals with beams. Having said that, beams can be a great looking feature and I especially like exposed horizontal beams, but again, that doesn't scream modern farmhouse to me personally. I hope this helps you understand what I was talking about, but it is your house so do what you love.