I think the motion you describe is called a "Dutch Roll" (it's explained in the link).

I have mostly had this happen in the Captain Sim 707 because the altitude and speed a controlled manually by me. So if I get too high and too slow, the 707 begins to rock from side to side. The only fix is to descend to thicker air (2,000 - 4,000 feet lower)!

I can't tell you why the stall warning horn did not come on. I have only had this happen the one time (explained in a previous reply).

Edit: Version 4.4 climbed very rapidly and due to some people saying it was unrealistic , it was changed for the last two versions v4.5 & v4.6. At least I think that's why it was changed. Someone can correct me if I'm wrong.