The Tag Grand Carrera at the top of your post is indeed automatic - it looks identical to one a friend picked up for me in Shenzhen a few months back. Was wearing it yesterday - not the highest quality rep (the GMT function doesn't work - it just shows the same hour as the hands), but good-looking for a cheapie and on a surprisingly good strap. Mine keeps excellent time, but the power reserve isn't great.
The BR02 will only take a BR strap. The rubber strap on mine is actually very good, but it was quite a bit more expensive than the one you've pictured (mine has a high-beat movement - another Shenzhen special), so I can't vouch for the quality there. Leather straps are available, but you'd have to buy gen or from somewhere like altstraps, so the strap is gonna cost more than the watch I seem to recall seeing 02s on leather somewhere - possibly Lawrence. Might be worth a look.
From the watches I've bought and/or seen, the low end for something reasonably decent is $50, and only a fairly tiny percentage of those are actually worth having. Ironically, I'm often more careful with what I buy when spending $50 than I am when spending $300, as a lot of the really cheap stuff is just street junk.
By the way - cheap autos with faux chronos can be unreliable as the extra complications to power the subdials (usually hour and day) are not terribly well made - I'd stick to non-chronos at that kinda price unless you're buying *cough* quartz *cough*.