"Accept our betrothal gift!"
"Yuinoohin" is a gift consisting of several items, including money, given to celebrate an engage-ment. The word "Yuinoo" can also mean the ceremonial exchange of such gifts. "Surume" (Dried squid), "Kombu" (Seaweed), sake and Tai (red snapper), items typically included, symbolize good luck. Orimono (hand-woven fabrics) are traditionally used to make the bride's wedding dress. The degree of formality one wants will dictate the quality and amounts of these items one will include. Such ceremonies were traditionally conducted at the bride's home, by messengers of the groom, but nowadays are held in hotels or wedding halls, when they are held at all.
"It's dark! I'm scared!"
This is simply a reminder of Mendou's deep-seated fear of the dark, which usually (but again, not always) goes hand-in-hand with his claustrophobia.
"There's a lot of things that I don't usually get to eat. This is pretty good."
This is a reference to the difference between Mendou's lifestyle, and that of people like the Moroboshi's. The things that he would have in his nabe are not the sort of ordinary things that people like the Moroboshis have in theirs, and he finds the change of pace interesting.
"I'll get some snacks to go with the tea."
"Chagashi" is, literally, tea snacks, the sort that one might typically have with Japanese tea, though not green tea, which is associated with the tea ceremony. Bancha or Sencha are the most common teas for chagashi.
"Why tag?" "It's what the Oni people do at fateful times such as this."
This line is one of the fundamental puns on which the series is based. The Japanese name for tag is "Onigokko," which literally translates as "Game of the Oni," and is thus logically the game that Lum's people, the Oni, would play. The significance of grabbing an Oni's horns stems from a Japanese myth which says that an Oni must grant the wish of whoever manages to do so. This myth in turn stems from a tradition in the Kansai region of Japan of lopping off the horns of all but one or two bucks in a herd of deer, to control breeding. Ataru's failure to be specific about the wish he makes on Lum's horns in the first story of the first episode of the TV series sets the stage for everything that follows, and in this film holding Lum's horns will enable Ataru to save everyone from darkness, thus granting him another wish, sort of. For more information, see Urusei Yatsura TV Set 1, Episode 1, Story 1, "I'm Lum the Notorious!" for the original game of Onigokko which comes more or less full circle in this story.
"Just say it! Which side is just?"
The banner in this scene reads "Tomobiki-cho choonaikai," which perhaps best translates as "Tomobiki Town Community Association," more or less a local citizens' action group. There is typically one in each section of a town, which gets together when members of that town need to do something for that town, such as cleaning the drains. They also make sure that information about government and other local affairs gets around to everyone in the area. Such groups are most active in older and more rural areas, where people are more likely to know one another, having lived in a given place for years at a time.
"Mendou SXR-7000-Tako, come on out!"
This scene is a parody of every giant robot series, whether manga, anime or live-action, to come out in the last 25-30 years. Also, the word "tako" means "octopus," which is the crest of the Mendou Family. Furthermore, when the Tako raises the ball with the Mendou crest on it, it is parodying yet another staple of modern Japanese pop culture, a long-running Edo-period samurai series called "Mito Komon." For more information, see Urusei Yatsura TV Set 4, Episode 14, Story 27, "Mendou Brings Trouble!"
"Momoe, I love you!"
In all probability, this line is a reference to Yamaguchi Momoe, who was Japan's most popular entertainer when she retired around 1981, to get married and settle down, at the ripe old age of 21.
The last note
Despite what you might think from the title of this film, the Urusei Yatsura gang will return in the 6th and latest UY Movie, "Always My Darling," which celebrates the 10th anniversary of the series.