I am a cell and developmental biologist and my research focuses on early development, so your question is right up my alley. My guess is that you and your students already know the answers to most of your questions once you think about the differences based on the kind of "twin" you are talking about.
First, we classify human twins into two major groups. First is the monozygotic (one egg, or "identical") twins and second is the dizygotic (two egg or "fraternal"). The word "zygotic" means from a fertilized egg and "mono-" refers to one, "di-" to two.
As the name implies, fraternal twins are the result of two separate eggs being fertilized by two separate sperm Thus, the twins share the same mom and dad, but not absolute genetic identity, so they are really just siblings of the same age, carried in the womb at the same time. This is the most common form of twinning in humans, and it is really not a formal type of "twinning" as defined by developmental biology.
However, identical twins are formed from a single embryo (the result of one sperm fertilizing one egg) whose cells (very early in development) somehow dissociate or split off from each other. In this case, the resulting twins have genetic identity and so look very similar (and are always the same sex). Identical twins occur in roughly 0.25% of all human births, so it is pretty rare. About one third of identical twins have complete and separate chorions (the "sac" that surrounds the embryo) which suggests that the separation of the small mass of cells occurred before day 5 (day 5 is when that sac forms). The other two thirds of identical twins share a chorion, meaning that their separation occurred after day 5, but before day 9. If the embryo "splits" after day 9, there is a good chance of conjoined or "Siamese" twins forming - the result of an incomplete split. There are some super rare forms of birth defects where one monozygotic twin fuses back with the other monozygotic embryo.
It is extremely rare to have triplets that are the result of monozygotic splits. Triplets or multi births are usually the result of multiple eggs being fertilized.
Interestingly, some animals normally reproduce by true "twinning." For example, female armadillos produce a single egg that is fertilized, and then, at the 4 or 8 cell stage of development, each cell (called a "blastomere") separates and gives rise to a complete new embryo and then baby armadillo. So all of the babies in a litter of armadillos from one mom are genetically identical.
Current census reports suggest that the incidence of monozygotic twinning has remained about the same for the last 50 years but there is some interesting data emerging about "hot spots" of fraternal twinning in the US and western Europe over the last 20 years. No one knows why this might be increasing, but one possibility is that more women are taking drugs to increase their fertility (in anticipation of perhaps IVF), and thus are ovulating multiple eggs in each cycle. the verdict is still out on whether there is increased rates of dizygotic twinning in humans and especially any causes.