Not really, sadly. In Mandarin, the old/literary kanji is 犬. 狗 is what we all use today.
The left-to-right rule is for components, usually units that you can easily isolate and which have a specific meaning. In other words, they’re usually for things that are listed as usual radicals in actual dictionaries. Also, the kanji concerned has to have a clear lefthand component and righthand component (or three vertical side-by-side components), which is not the case here.
止 is a kanji in its own right, as you probably know. It’s used to mean ‘stop’ now, but traditionally, it was the shape of a foot. The bottom half of 歩 is the same thing, but stylised and transformed differently. It’s two feet, like footprints on the sand, hence the act of walking, or the footsteps themselves. Here are some older forms:
Since we have two components here in a top-and-bottom arrangement, we write the top half first. As for everything else you mentioned… central vertical strokes that end in a flick or otherwise link to other strokes tend to be written first and help to centre the kanji. They’re written last when they’re long and go right through the entire kanji without needing to link up to anything else, like in 中. And no, little things don’t always come after big slides, especially if the slide doesn’t link back to those little things. You’ll notice that when dots/dashes are written last, they’re usually preceded by a stroke ending in a flick, like in 心.The kanji flows much better that way.