Road chainring rubbing on frame
I don't think you are missing anything. I think these frames were intended to be used with mountain bike triple chainrings. The fully built up version that Trek offers comes with Alivio/Sora 3x9 with a 48 tooth big ring.

If the 52 an 36 tooth rings are just hitting the stay you may find a 50/34 pair provides enough clearance (A 50t ring is 4% smaller diameter, 34t is 5% smaller). The 105 crank has a BCD of 110mm and 50/34 rings are available for it. This would be the cheapest option I think. You should be able to sell unused 105 52/36 rings and largely offset the cost of the new ones.

If a 50/34 does not look like it would provide enough clearance then there are sub-compact 48/32 cranks available (8 and 10% smaller diameter) and I thikg you should be able to find one that is compatible with the Shimano BB. Again, you can sell the unused crank to offset the cost.

Update to add info on chainring compatibility:

Chainrings with a difference of less then 16 teeth (52/36, 50/35, 49/32 etc) will work well with a Shimano road derailleur. A potential problem with small chainring pairs is lowering the derailleur enough. This can be a problem with braze-on mountings but is generally not a problem with clamp on. The limiting factor for chainring size on a crank is the bolt circle diameter. For compact cranks with a BCD of 110mm the limit is about 34. This is for the smaller ring obviously. You can mount a 48/34 pair but then you might have a problem of the high and low gear ranges being too close together. 

With Shimano cranks you have to be aware of the differences between the groupsets: bolt pattern and the shape of the ring where it meets the crank body makes inter-groupset compatibility poor.

I think you easiest route would be to sell the 105 crank and get a sub-compact one. I note there is a Tiagra 48/34 option which will fit in your BB.