A phone as a home 4G modem/router? I live in a rural area where traditional cable, fiber, and even DSL are not a possibility.  I am using a weird wireless system a local ISP set up for this area, but its flaky, slow and expensive.  The only thing it really has going for it is there are no data caps.  

I can get 4G signal for T-Mobile, however (and Verizon, but their data caps drove me away), and I'm considering making my home internet connection consist of a dedicated Android phone on a T-Mobile 4G unlimited contract, using its WiFi hotspot.  (I would probably set up another wireless router for the actual LAN so I'd have good coverage of the house, but it would use the phone's network as its uplink.). I would use a phone rather than a USB modem to take advantage of T-Mobile's unlimited data package.

Is this a bad idea?  Would the phone fry if I left it on and broadcasting its WiFi hotspot 24/7?  I have an N5 but I've never used the hotspot for an extended period before - are they flaky?  Would it drop IPs or anything like that and require reboots?  Would T-Mobile decide I'm breaking a rule and throttle me down?
That's not a bad idea at all. I have Verizon Wireless and I do this all the time. That's what tethering is for, and the USB option as well. 4G is fast, and for what it's worth, based on what you've said, it seems like a fantastic idea.

To answer your second question and forward, no - -  your phone wouldn't fry, providing you got a phone with good specs. 

As far as I'm concerned, they have been more reliable than regular average routers. It's LTE after all, so it's not a surprise. Can't speak about the throttle, that depends on T-Mobile and their terms. I think they do throttle, but it's not a huge difference. 

I'd recommend talking to someone from T-Mobile and asking them what device has best specs and battery power. Also, you can get a data monitor to monitor all traffic and block / allow certain connections via incoming addresses or devices. Those apps are available on Google Play so let me know if you need them. 

I have a family member who uses T-Mobile tethering  amongst six devices at home, two of which are a PS3 which plays Netflix and it's not slow at all. Randomly given of course, every server has ups and downs.