I can't comment on the first part (even though it sounds insanely dodgy), but I can definitely refute the second half. The answer is very simple: the Romans were fighting a bitter war with Persia between 602 and 628, and from 608 onwards they were losing very badly, to the extent that all Roman territories in the Near East had been lost by 621. This was of course when Muhammad probably began to solidify his following, so it really doesn't make sense - the Romans simply had bigger issues to deal with than an Arab warlord active in an area they had no strategic interests in.

Still, I'll admit that there are several ways his or her argument could work, if it is very nuanced and it is phrased very carefully. There was for instance a Roman-backed Ethiopian invasion of Yemen in the early sixth century, which drew Arabia firmly into the great game between Rome and Persia. Their influence very plausibly may have created a situation where social, religious and political conditions were suitable for a new religious movement to be created under a charismatic leader. There were also instances when Arab confederations serving the two empires, the Ghassanids and the Lakhmids for the Romans and Persians respectively, fell from their imperial masters' favour, which must have played an important role in shaping the mindset of contemporary Arabs. Muhammad for instance allegedly reacted positively to an Arab attack on Persian forces (still consisting of Arabs, but now they were under direct Persian control) in the 600s. So in this atmosphere I guess it is possible to make the case that because of the Romans' horrible treatment of their clients, disaffected auxiliaries were on the look out for a way to get back at the Romans, which might then have influenced the military direction taken by the nascent movement led by Muhammad.

However, these are all 'ifs' and 'maybes', and comes with all sorts of qualifications and were most definitely not the singular cause of the rise of Islam - there were many, many factors behind its rise and simplistic explanations are all inevitably wrong. I hope this is useful; if you can remember any more details from the workshop you are not sure about (on this topic at least) I'm happy to try to answer them :)

Regarding the first part: as others have mentioned, Joshua/Yeshua was and is a Jewish name, and was an extremely common name, too. It was not a title, and would be extremely confusing as a title, given how common the name was. Christ/Christos, however, is a title. It is a Greek translation of the Hebrew Messiah/Moshiach, which means "the anointed one". This is a reference to the Jewish belief that a great mystical leader will emerge, and the anointing is symbolic of his important role as a savior. But this was not a title that was actually given by the Jewish people collectively to any person as part of a revolt or anything. It was rather a way to refer to the hypothetical person who would come to fill that role. At various times, small groups of Jews have applied the title to some man or other, because they believed that man was the anointed one, while others equally passionately did not believe that person was the messiah. See Shabbtai Zvi and Menachem Schneerson, for examples. But it wasn't some title that Jews in general used to refer to a leader of one of the revolts, which they then passed down to others. The way this guy phrased it, it sounds like he thinks it's like "Chairman of the Joint Chiefs of Staff" or King or something.