Yes and no.

Modern racism is a post-Classical concept, it does not appear in surviving Egyptian, Greek or Roman literature or other accounts.

Xenophobia, bigotry, sectarianism and extreme nationalism were all amply represented in Classical times but the racism familiar to us today was pretty much unheard of.

Were marriages between someone from the Middle East or Africa and someone from Europe common?

Yes, there are many sources describing or depicting marriages between Romans and non-Romans, both in Rome and in the provinces, from common legionnaires to senators and emperors.
As to the possible stigma against such marriages it is difficult to say exactly how the Romans may have felt about any given ethnic group at any given time, but generally speaking, they were more concerned with culture, power, wealth and social class or standing. A Roman, and one of Latin descent, stood above all, whereas a barbaric appearing Gaul might be considered contemptible. The poor were generally frowned upon as much as happens in any society, but Roman citizens were still more respected than any barbarian culture. Greeks were often accepted/admired (depending on the person or group of course) above various 'barbarian' cultures because of many similarities. However, evidence seems to show that a ethnicity played little part once that group was Romanised. If they looked, acted and behaved like a Roman, then to most, they were a Roman. Non-Romanised foreigners were certainly subject to some disdain.

Were Julius Caesar and Cleopatra considered an "interracial" couple?

No, because the concept was unknown, add to that Cleopatra was a Ptolemy, a Greek family originally installed after Alexander the Greats conquest of Egypt.

But the marriage was extremely unpopular with the Roman senate. Calpurnia (Caesar's official wife. he wasn't married to Cleopatra under Roman law) was the daughter of a prominent Roman family and while like a good Roman woman she was subservient and demure, Cleopatra was powerful and outspoken. Caesar had many enemies in Rome by this time - chief amongst whom was the orator Cicero. Cicero declared Alexandria "the home of all tricks and deceits" and openly admitted that he despised Cleopatra. He was of the opinion that women were intellectually weak and should always be under the power of their male guardians so he was greatly concerned about Cleopatra's influence over Caesar. Rumours began to circulate that Caesar planned to move the capital of the Empire to Alexandria and senators began plotting how they could reverse Caesar's reforms.