Alkaline means "not acidic". Basically, if you test your pH (acidity or alkalinity) in the morning (urine stream) and it's low (lower than 5.5), then your body is too acidic and you need to lower your night time acidity (i.e raise your pH). You can do this with calcium and magnesium supplementation -- or, for that matter, a number of minerals (including magnesium and calcium) can do it.
Why are you on a completely Sugar free diet? If you're avoiding processed Sugar by choice (and not, say, diabetes), then whether or not you use apple juice is up to you. Freshly juiced apples are excellent.
Maybe your issue (and please note that this is a wild guess, backed by no scientific or medical evidence whatsoever) with the magnesium is that you've taken enough to loosen up the bile duct area some, which makes it easier to move stones around, but you haven't loosened it up enough to allow the stones to pass. If this is the case, you could test it by tripling the magnesium you take. (I'd avoid the antacids, however.) Don't triple the Epsom salt; but you may triple the magnesium oxide (or carbonate, gluconate, aspartate, citrate, or... whatever "supplement" form you can find).
Don't expect it to be a one-shot thing, however. You might need a lot of magnesium for awhile to help restore some balance if you're particularly deficient.