I have almost decided on the gearing for my touring bike. The rear is a Sun Lite seven speed, freewheel with 14-34 cogs. Wheels are 26 inch mountain bike. I cannot "upgrade" to a 8/9/10 or 11 speed rear.

I am thinking for the riding I do and plan on, including a cross country ride pulling a trailer, I should use chain rings of 38 (or 36)/32/22.

I am not racing, I do not use the 48 tooth big ring I have now, 90% of my riding is in the middle 38 tooth ring, the rest in the 28 tooth granny (Bio Pace rings). I know I will have to change to a compact crank in order to get the ring sizes I want; most of the sizes I want are not made in a 5 bolt pattern anymore.

Will the 38/32/22 rings give me a useable range of gears for touring? I expect the trailer to have close to 100 pounds in it, plus a little in the panniers.

It sounds like a good range of gears to me. My temptation would be to use a slightly larger big chainring -- maybe a 42 instead of a 38 -- but gearing is personal. Ultimately, what works for someone else won't be ideal for you. That your lowest gears are as low as they are is a good thing in my opinion. I have a similar gear set up myself -- my low and middle range are almost the same as what you are proposing, my high end is somewhat higher -- and it works great for me.

As acantor said, I would also be tempted to have a slightly larger biggest chainring. I've got a 44/34/24 chainring setup, and I understand what you mean about spending most time in the middle ring - but sometimes it's nice to have the longer top-end ratios for pedalling along slowly on a descent or with a tailwind (they DO happen sometimes). Or when I'm not loaded (the bike, that is!) after setting up camp and just heading to the pub for dinner or something.

Sheldon's gear chart shows you going about 16 mph @80 rpm in high with the 38, flats on a smooth road and a good day. With any decent tailwind, unless you're a true spinner, means you're likely to run out. Might want to bump the top ring up to 42 to take better advantage of those 'gift from God' days.

I put thousands of miles on a 14-34 ,6 or 7 speed 126 axle spread..
110/74 crank set 50 (95"high, 700c wheel) 40 or 38t , & 24t ganny cog.

The 38 is a very low especially when used with a 14 tooth freewheel. Why not go with a 44? You probably won't even have to change the crank. Bikeman (and others) carry lots of rings in lots of different bolt circles.

On a side note: What possible reason do you have for carrying 100 pounds in a trailer and then adding more in panniers? Trim down your load! I pack heavy and even I don't lug around 100 lbs of stuff.

Packing list... or it didn't happen!

I don't do packing lists I ride. I carry and if I don't need it I send it home.

^^ didn't mean you cyccommute, meant OP's 100+ lbs load.

thats a crapload of stuff.

My tent and most of my other camping gear is heavy, and I cannot afford to buy lighter.

6 person 3 season dome tent, cast iron skillet and porcelain on steel 2 qt. pot (all my skillets are cast iron), Coleman 2 burner stove (Coleman fuel type) laptop and required power supply and cables, Mag Light flashlight (choice of 3 or 4 D cell model) a weeks worth of clothes, (I do not own any cycling clothes), a couple tins of chewing tobacco, 10 or so (ok, maybe 20) boxes of Lipton Energize tea, spare "thorn proof" tubes for bike and trailer (unless I get a pair of airless tubes for the trailer) 2 tires each bike and trailer in case of cut sidewall or tread, spare brake and shift cables, a bike's worth of spare brake pads, lube for chain, small tube of grease for hub bearings (God willing, will not have to grease bottom bracket again for 10,000 miles or so) first aid kit (might omit that) extra batteries, kerosine lantern, kerosine or lamp oil, maybe a old fashioned non-electric coffee pot (if I can find one at a second hand or antique shop) and 2 pounds of coffee, dry creamer, sugar, sandwich fixin's, and a couple gallons of water. Can't think of anything else right now. Its getting late and brain went to sleep.