Even on a cheap hub, your lockring shouldn't come loose. Cog threadings and lockring threadings can be of several different types. In cog/hub threading, you want to match BSC to BSC (that's British Standard) and in lockrings, your choices are typically Shimano (another BSC threading), Campy (aka "Italian," same as Phil Wood), and Mavic (aka "French"). You may be mismatching your cog and hub, and your cog won't go farther by hand because the threads are jamming up; under pedal power, they move a little farther. Your lockring can only come loose if your cog isn't down all the way or in rare circumstances if the lockring isn't the right thread (but the mismatch is pretty obvious).
Oh, and some hubs have threading that's wider than the width of the cog, so your cog goes on just like it should, all the way, but there's still a lip before you drop down to the lockring reversed thread. In this case, you should end up cranking the lockring hard against the lip, which will hold it in place anyway, although the cog may loosen up a bit. Inspect for this and if it's the case, get a bottom bracket spacer ring (they come in various thicknesses so find one that fills the gap) or try another cog. As others have said here, you have a budget hub and don't want to invest much in it, but it should work all the same.