Is it true that vaccines can cause the conditions they are supposed to prevent?

No. Most vaccines are made of "killed" or inactivated components of the viruses or bacteria that cause a disease. Some vaccines are weakened "live virus" vaccines, which do actually give you a very mild form of these virues (like measles/mumps/rubella vaccine, chickenpox vaccine and rotavirus vaccine). The viruses are weakened so the do not cause severe illness but may cause a fever or a mild rash.
No. For the most part, no. You cannot get polio, pertussis, or measles from the vaccines. Vaccines do have small side effects and occasionally bigger ones, but no, they don't cause the disease, with 2 small exceptions. The varicella vaccine can occasionally cause a few scattered pox, and, very rarely, illness, and the flu mist in theory can cause flu, 0.0038% chance.
No. All of the routine vaccines are safe and none of them cause the horrible diseases that they are supposed to prevent. All of the vaccine preventable illnesses are potentially fatal so get vaccinated.
No. Not any more. The old live polio vaccine and the smallpox vaccine, which did cause disease on occasion, are no longer used in the US - one for over 30 years, one for nearly 20. Vaccines can, in very rare cases, fail to prevent these conditions.