First of all, given all the schmucks out there trying to make a movie that have never done so before (even in L.A.), I commend you for looking into the legal requirements and checking out SAG guidelines.
Yes, you must have a set teacher present at all times. In general, it's required there be four hours of instruction on any normal school day the child actor works. On non-school days, no instruction is required, but the set teacher must still be present. The number of work hours permitted in any day, and any week, is tiered, so it will completely depend upon the exact age of the individual as far as what those guidelines are. You're best bet is to refer to the SAG website for that or your local Dept. of Labor website.
As far as Alabama law, I don't know from my experience, however, in general you must obtain work permits for the children, yes. A guardian will have to sign off on them, as will a producer. Check out your state's Department of Labor website for more information.
Since it sounds like you want to do this the RIGHT way, and again, I really commend you for that (SOOO many slimey people in film), I would also check out the process for becoming SAG signatory for your project. It will add legitimacy and help if you wish to distribute this at some point. It's a pretty simple process, and doesn't cost much, so I would look into it.
And thanks for a small chuckle. I don't know what your film experience is, but "semi-low budget," aside from being a term that doesn't really exist, means something in the vicinity of about $5M where I am. I'm guessing you're not making this for $5M, but if you are, let some of your investors know I'm producing a ULB project right now that needs to complete funding. :)
I'm a film producer.