No, the stored food - or anything else except livestock - has no time-limit or an expiration date.
I searched around for tips and found a person who did precisely what you suggest: use a trading post as an additional storage barn. I tried it myself, and it works like a charm, especially with food. So yes, do use it as an additional storage barn.
There's even one big plus thing to it, which also answers your last question: nobody but the trader will carry things around, and the trader will ONLY do so if you order them. (Meaning you have to set the desired amount of a certain food to more or less, depending whether you want the trader to bring stuff to the trading post or to the market.)
How is this a plus? Well, it allows you to micro-manage goods better.
For example, when there's a surplus of food, you can order a trader to bring a certain amount to the trading post and your people won't hoard it or eat it, because they have no access to it. And if there's some disaster or just a famine episode, you'll be able to tell the trader to quickly bring it back to the market, and voila, starvation avoided. Also, keep in mind that things in the trading post, much like people's inventory, don't show up in the tab with stored goods. So you can have a few hundreds of fish stored in trading post that won't show as food you have until you send them back to market.
I've been using this tactic for a while, and I think it's wonderful. :) No more starving hoarders.