Thanks to Billy and Ernie over at the Yesterday's Trucks forum, we have an answer to the "What kind of truck is this" question, yay!
Billy answered with: "[S]ure looks like a model A to me... somewhere between 1928 to 1931."
Ernie verified that and narrowed it down more with: "It's a 28-29 Ford AA. The pickups were model A's but the big trucks were AA. The 30-31 Fords don't 'scoop' in at the cowl like a 28-29 does."
Ernie also provided some other fascinating tidbits thusly:
"It's also interesting that 28-29 trucks share doors with Model T's, I guess Henry had some left over."
And when asked when they stopped making Model A's, he responded with:
"The last year for A's was 1931. The classic 1932 Ford came next. They came in 4 cylinder versions - the model B - and V8 versions, the famous Flathead! They are a little more rounded, and use the "French Curve" design extensively in body lines, dash, headlight bar, etc...If you happen to have one out behind the barn it's worth some money, it's the holy grail of early Fords. The 33 and 34 trucks were very similar to 32 but the cars got a thinner, laid back grill and smoother body lines."
Great answers, guys! Thanks a bunch!