Why does our toilet keep blocking and then slowly draining

Your first step should be to try to divine where in (or outside of) your house the blockage is. If your outside drain is overflowing you know the blockage is downstream of that, so if that drain is teed off the main waste line coming out of your house, you know you've got a block between the drain and the sewer or septic tank. If that's the case the most likely culprit is tree roots growing into the pipe, a problem that's not uncommon if your pipe is old and made of clay. If your house is newer you may find a cleanout port outside you can snake to try to clear any blockage.

If the outside drain is plumbed into your basement or crawlspace then you should look for cleanout ports down there too. However if all the drains in your house work slowly at the same time, then your problem is likely still outside the house.

Given the seeming interconnectedness of your issues, my gut tells me you've got a blockage that's outside the house, and it's affecting everything. Your toilet is probably stopping up because of the low water flow -- if the liquid's not flowing out of the toilet quickly enough there's not enough force to effectively push the solids out, and you get a block.

You have a 3-year old (deep sigh) - odds are pretty good it's not a 2-week old subscription card (ought to be pretty much mush by now) but some other object likely flushed more recently.

My slow-toilet (unreliable) magic trick is dishwashing liquid, which sometimes (but not always) lubricates things enough to make plunging far more effective. Just a little is all it takes, and using vast amounts won't help more. A cup of bleach overnight sometimes also helps, presumably from breaking things down a little and freeing some sludge. Verify that you can (or refresh your memory on how to) unlock the bathroom door from outside, and lock it while the bleach is in to keep the 3-year old out.

Pouring in hot water can also help, but overdoing that could damage the wax seal, and then you'd be pulling the toilet - which is one more option for desperate cases, albeit one most people prefer to avoid. But if there's something like a doll's head or golf-ball down there, a snake may not grab it, and then pulling the toilet is the only thing that will work. I would agree with @Keith Hoffman that it's nothing to fear, and the tools required are far more general-purpose than a snake (wrenches, mostly.) It just crosses into the "icky" factor that makes plumbers so much money ;-) - face that and save. Sponges and rags are an alternate way to get the water out, and remember to have a plastic bag stuffed with something (other bags or paper work) to plug the hole while the toilet is off (and don't forget to take that out before replacing the toilet.)

The main reason (AFAIK) to strictly avoid Drano® (et al) in a toilet is that you have a MUCH higher chance of causing yourself (or the 3- year old if she gets in there while it's in) serious chemical burns from splashing in a relatively open toilet, rather than a more contained sink drain. My personal experience from sinks is that it usually doesn't help anyway, but does make the next plunging session much more worrisome.