Does my previous travel to other countries affect my ability to get a U.S. visa?

Your international travel (or visas to other countries) will be taken into consideration during the visa interview process only to the extent that it demonstrates your ties to Ukraine (that is, your ability to financially support your trip and your intent to return to Ukraine in a timely manner). The political relationship between the countries you've traveled to and the U.S. has no bearing on whether or not you are eligible to receive a visa.