Container Help

I won't have a full answer for you here, Andrew, but I would suggest that you may want to limit your scope of movement to one object - your spider.

To do that, you should nest all your legs, eyes, etc. inside of a movie clip symbol that you create in Flash. All your parts, too, should be movie cliup symbols (then give them instance names so you can access them to affect x, y, scale, alpha, etc). The movieclip symbol for you should be a manually created object in the Flash program, versus other ways of doing things that are completely code-based. This is that same as the "container" you mention. (movieclips and sprites are the main DisplayObject containers in AS3 that can hold other items within them).

Your movieclips' instance names could be:
spider
(and then the instance names of items inside the spider symbol)
- leg1, leg2, leg3, etc.
- eyeLeft, eyeRight

Then, in a basic sense, with the spider on the stage you can set a couple basic things:
spider.y = 200;
spider.eyeLeft.rotation = 45;

But, you also have to deal with bringing your spider onto the stage if it isn't already. Start with getting familiar with creating the physical art, converting that to movieclip symbols, nesting them inside a "container" spider, and then look into how you bring things onto the stage and move them with AS3.

You can group several movie clips into one container by selecting them all on the stage and choosing Modify - Convert to Symbol, then use the caurina tweener class with the nested movie clips from the main timeline. So for instance if you have a movieclip container called spider_mc which contains another movie clip called eyes_mc, you can tween it's alpha like this from the timeline:

Tweener.addTween(spider_mc.eyes_mc, {alpha:1, time:1});