1. I can't answer that, except to give my opinion that if you're growing in plastic (like I do), then you accept whatever leaching occurs. I've read here of people getting foodsafe plastic buckets that previously contained cooking oil. I would go with the ones you get for free unless they had something toxic in them.

2. Potting mix is probably not what you want, as it may not drain well enough. There are threads on here on how to mix your own for better effect. If you were were going to pick between the two you listed, I would suggest not filling the bucket with the seed starting mix. Use the seed starting mix to start from seed if you're going to transplant it from your seed starting trays into somewhere else (like into your bucket), or if you're going to direct sow you could make a little pocket of seed starter in your other mix if you were so inclined.

3. I would say it doesn't matter much if you're just starting out. Find something that you'll eat, for instance, lettuce. Look on Park or Burpee that you mentioned or High Mowing Organic, or even just a big box store like Home Depot, Lowe's, etc. Look for organic packets. Not all the veggies will need to be dwarf or container sized varities, for instance you could grow some green beans or lettuce or radishes or carrots or spinach or ... without too much trouble. Most of the larger plants that you'd need dwarf varieties of will not appreciate a 5g bucket for it's feet anyway (too small).

The important thing is to just start where you are - if you've got soap buckets, clean 'em out and use em. Use the space you've got, get the plants in full sun, and experiment. Keep in mind, you will need to feed these plants directly (not using compost or something that requires a biosphere to process into usable nutrients). You can always get better things later, it will never be perfect.

Grow something you'll like to eat and you'll continue to be interested in gardening.

I'm not an expert at this by any means, but I saw no one else had commented, and I wanted to point you in a reasonable direction.

You may also consider starting with some plants instead of seed, unless you're really interested in watching it develop (it takes patience to watch it grow from seed, especially if you're excited and interested).