Can someone tell me what hose/filter I would use for connecting this to a 17-pound gas grill propane tank? I'd like to order the right accessory.

There's a couple of choices. The F273704 hose by Mr heater will work alone (no filter needed) and is likely the cheapest way to go overall. Otherwise hoses like F273701 or any other that goes from a tank to "disposable tank" connection can be used along with the F273699 filter. The reason for the "filter" is that most hoses are made with rubbers and oils in the rubber get "squeezed" out under high pressure and mess up passages in the heater, so the filter takes out those oils (yet the 704 hose is made of "oil free" materials to not have that issue). Some ask "can I use the F271803 hose/regulator with this?" where the answer is "No". This heater doesn't have the QD port for low pressure gas, only the "Big Buddy" models have that port. FYI I even tried plumbing gas from a regulator to one of these and it lit, but with it set to "high" had a faint glow as if normally set to "low" (hinting a lack of gas getting to it). So to get it to bright orange you must feed high pressure gas letting it's regulator do the job.

How long does this heater run on the propane bottle that fits on the side?

Depends on your setting of "high" or "low". I burned a 1 pound tank in 3 hours on "high" and on "low" it went about 5 hours. Some others saying "8-10 hours" sounds like they are talking about another model (the "Big Buddy" with 2 burner panels) that sounds high to me since this only uses 1 burner and 1 tank with just "high" and "low" (they say "Med" as well as "both tanks" as hints of wrong model). Others point out some things like in theory 1 gallon of propane gives 90,000BTU where a quart should be 22,500, yet this running for 3 hours and supposedly giving 9000BTU would seem to be 27,000 BTU (where the numbers don't match up). I don't have anything to measure actual BTU given, so wonder if the 9000 label isn't a bit high (or perhaps the run time is a bit high?) Perhaps even the one quart being assumed to be 1 pound is slightly off. Not sure how "Exact" you need the numbers to be, but this should give you a reasonable estimate.

is this a good product in the midst of a power outage?

I'd be OK using it for such a situation. I've even done a drill pretending to have a 24 hour outage a few times and running one of these with freezing temps outside to see how it goes. It kept things good for me (and I only have electric heat, no gas for water heater or such). Note that these will use a 1 pound small tank in 3 hours with it on high, so depending on your needs you may need 8 pounds for 24 hours run time. Many want to connect to a BBQ tank that can be done with a hose like the F273704, but note it's against fire codes to take any tank over the 1 pound size indoors. Some may decide "I'll do it anyway in this emergency", but no one wants liability for saying to do something illegal. One of the biggest fears is "will the fumes from burning gas be like running a generator indoors and kill me?" The answer to that is "No" running this is more like running a gas stove burner (no one dies from cooking food on a gas stove, do they?) That said, they do suggest cracking a window an inch for some venting, but many say "That's letting your heat out and seems stupid". So it's up to you to do some experiments and see what works for you depending on the size of your place etc (which only you know of). I set up a CO alarm nearby while doing my drill and it never went off. I did note a slight "smell" of the burned gas when I went out for the mail and came back in, but it wasn't overwhelming. There's a few other things to be mindful of. Checking for gas leaks is always a good idea. The heat can be a fire source for things nearby. They even warn in the manual "May cause hardwood floor fading or vinyl damage" to show what it sits on matters (some heat will go down as well as up). The tanks will have "frost" on them as they get emptied and have water dripping off, so don't be surprised to see that (even on the BBQ tanks). So there's lots of considerations, but it can do a reasonable job providing heat as it's supposed to.

Why would you need a filter when you use 20 lb to 1 lb converter hose

Most hoses have rubber inside where rubber contains oils to make it flex. When put under high pressure, those oils get "squeezed" out and join the gas stream into the heater where they damage it. So all hoses going to a "disposable tank" port need the filter (except the OEM F273704 hose made of "oil-free" plastics). So it's not "tank crud" that the filter is there for, it's the oils from rubber.