Why is he teaching a Judo class? Is this a school club and he was the most qualified or is this a private dojo?
First the short answer is Judo does owe its origins to jujutsu. The founder of Judo was trained in jujutsu (a sort of catch all term for Japanese empty handed martial arts) prior to creating his own style. This happened in the 1880's so truly Judo is young compared to jujutsu and Brazilian Jiu-jutsu is in its infancy. His first students were trained in jujutsu and possibly Aiki-jutsu prior to becoming Judo men. Eventually a Judo stylist went to Brazil and wound up teaching several brothers of a family named Gracie for a time. The Gracies would eventually come to refer to their art as jiu-jutsu. Gracie Jiu-jutsu is a trademarked style in this country with Rorion Gracie holding the trademark. Brazilian Jiu-Jutsu/BJJ is what is generally used when referring to this Brazilian style of grappling. This liniage is why I have heard Judo men make the tounge in cheek claim that BJJ stands for Basically Just Judo. Yes the majority of BJJ techniques can be found in Judo books. The emphasis on ground work/ne-waza is the most defining characteristic but some Judo schools also emphasis ground work, I am thinking of Kosen Judo here. As far as sport/technique goes at this point the competitive rules of each sport are driving the differences in technique more than any real style vs style purity.
Because Jujutsu is sort of a generic term claiming what it emphasizes relative to Judo is sort of meaningless. The sensei could be correct or dead wrong depending on the Ryu of Jujutsu in question.
This is kind of pseudo traditional. Demonstrate, students mimic, no questions until you have done it 1,000 times. And of course once you have done it 10,000 times (10,000 being an analog to infinity here) you will have transcended the technique and reached elightenement. I somethimes miss this, although I think there are better ways.
Not "traditional", of course I feel driven to point out that most of the rigid formality in Japanese martial arts was instituted in gearing up for WWII. I am not so sure the uniform thing matters all that much. The cut is a bit different. Karate gi's tend to rip easier. The big one is the collars tend to cut and tear your skin when you get lapel choked. If you want to wear a Karate gi to grapple with me I say fine. Just costs you more money and maybe a bit of skin/blood.
Ok, so that is settled. I would check to see if he was a sub or the only teacher though.
No way to know unless you watch/try. If you cannot go anywhere else because of geography you may have to dance with the gal that brung ya. Is all of this happening through on campus student clubs, or is this a Dojo? Are you paying? How much? I really want to know what the MMA club is teaching/training. Is it grappling, but no one has a belt so we call it MMA. Are they actually competing in amateur MMA matches?