This does suggest that a DSLR is what you need to get the type of image you want. That is perhaps somewhat more true of shooting in Alaska than other places...
Some significant advantages of the D7100 over a D5300 are the weather sealing, the longer battery life, more cross points for AF, and 2 memory cards. Of even greater importance is the pentaprism viewfinder vs a pentamirror. And possibly in the long run the built in focus motor which allows you to use certain older AF-D lenses as well as the modern AF-S lenses.
That adds up to the D7100 being significantly better for what you are apparently intending to do than virtually any other non-full frame DSLR on the market.
This indicates you are interested in landscapes and general protography, but not particularly in wildlife? If wildlife is in fact of interest, take a very hard look at the new Tarmron 15-600mm lens. And that is true for those with Canon cameras too, because while the Canon 100-400mm is nice lens, it just isn't the same as a 600mm lens.
Personally I would not really consider using the 18-140mm. The zoom range is too high. The 24-120mm f/4 is a fairly good lens, and with one of the other wide angle lenses for landscapes it would serve well as a walkaround general purpose lens.
For renting, as far as I know Stewarts Photo in Anchorage is the only "local" store that has rentals. Google them to get a phone number and their webpage, and check to see what they have.
I think because I am one who is often susceptible to buyer's remorse, I will feel somewhat guilty about whichever decision I make. But I think I will feel worse about potentially missing out on features in the D7100, which is why I may ultimately go it over the D5300. Why can't Apple just make a DSLR?
Anyway, yes I am primarily interested in landscape and general photography. Wildlife interests me somewhat but not as much. I would definitely have to rent both of the lenses you recommended as they are each over $1,000. I'm actually just thinking about renting a Sigma 10-20mm or Tokina 11-16mm, rather than buying them outright, as I'm not sure how much I would use them once I'm back home. I think the 18-140mm is a good general purpose lens, especially just starting out. The zoom range isn't as drastic as that of the 18-200mm or the 55-300mm.
Also, I see from your location that you're from Alaska! Nice website too! Have you been to Anchorage? I am very excited about visiting the Girdwood and seeing Matanuska Glacier.