Your post wasn't all that clear, so here are my assumptions:
- You have two switches, one in the rack and one near the house network patch panel
- You have one patch panel, near the "house switch"
Even though I am not expert on fiber cabling, I think I can say quite definitively that using DACs like that will not work.
If the distance is not too far, you could connect the switches directly to one another.
In fact if you have only one patch panel, why even bother with using it?
I am quite sure (but again not an expert) that you can get fiber keystones, like this one on amazon (mentions use in patch panels as well):
Amazon.com: Networx Fiber Optic Keystone Coupler - LC to LC Multimode Duplex - White : Electronics
www.amazon.com
Whether you should use RJ45 based 10G depends on the switches you have, but if you have SFP+ on both, I see no reason to use that unless you intend to use existing CAT6/6a/7 cabling.