For the sake of discussion, yes, mules are always sterile. But, as there are exceptions most all rules, fertile molly mules have been reported.
In most cases, the mule's baby is born either a full donkey or a full horse, depending on who the father was. (Mule + donkey = donkey foal, and vice versa.) In rarer cases, however, an actual baby mule (a mixing of donkey and horse genes) is born. We can not explain this. It's a wonder of nature.
One of these cases was Dragon Foal, a mule born from a mule mother in Asia, and Blue Moon and White Lighting, two mules born in the US to the same molly mare a year apart. (I believe the father, a donkey jack, was the same to both babies.)
And yes, mules can be used for embryo transplant. Their reproductive systems work perfectly well, so they can carry a baby, produce milk, ect. The only problem is that unfortunate chromosome count. Also, molly mules are famous for stealing babies from others! Some of them really, really want to be mothers.
A jack is an ungelded male mule, a molly (or sometimes called a jenny or jennet) is a female mule, and a john is a gelded male mule.