How does the tire warranty work on OE tires?

No, OEM's are pretty much a write-off. Typically I think, only tire defects are covered, and that is by the tire manufacturer, not Toyota. From page 29 of the US Warranty and Maintenance Guide:

I'd recommend Bridgestone Ecopia EP20 as a cheap LRR with decent tread life, they are Toyota's OEM choice in some regions, come in 195/65R15. Actually that might be the only size they come in. One downside, they are terrible in snow once you wear off about half the tread.

 

You should have received a tire warranty document from the tire manufacturer when you took delivery of the car--see what it says.

 

I went with Michelin X-Ice, in 195/65R15 (even though our stock tire is 17"), on steel Corolla rims. When new the X-Ice had a fine corrugation on the face of tread, and it barely wore off in 5000km use last winter. In short, they look to be long lasting.

They ran very quiet and the car felt planted, no bare pavement slippage (we barely had any snow, LOL). Mileage on par with our wider, low profile OEM, fwiw.

 

Your safety is more important than a small increase in fuel economy. The LRR specific tires are lacking in handling and traction compared to many grand touring and higher performance tires. Due to market trends, most of the "conventional" grand touring and performance tires have some kind of fuel saving technology in them, but obviously not as aggressive as the LRR tires.

I would get a set of the Michelin Primacy MXV4 tires. They are $116/each on Tire Rack, and there's even a $70 mail-in rebate on them right now. I've driven a few cars with them, and they are quiet, last a long time (60k warranty) and handle great for an all-season.