Could bedbugs cause any serious sickness?

Not likely. Just bites are more of nuisance, however, it the bite sites get infected, it my can cause serious consequences in the presence of other illnesses, e.g., diabetes, immune deficiency.
Not serious . Nothing serious, but some people develop an allergic reaction to a substance in the saliva of the bedbug.
Probably not. An article in the cdc's journal last year showed that bed bugs are capable of vectoring (transferring from one person to another) various kinds of bacteria. However, there is as yet no evidence that they actually do it; that is, nobody has ever gotten a disease that there is any proof was transmitted to them via a bed bug bite.
No. Most likely an uncomfortable itching of the skin, usually legs; not an internal problem.