This is called localization.

Information or brevity will always be lost in translation (and communication in general), so the translator needs to pick what information to convey and how to convey it. Sometimes it can be difficult to find a satisfactory localized translation, like translating a pun. Or if the translation has been localizing the Japanese name-honorifics system as whatever the characters in an English speaking country would call each other, but then there is a long dialogue in the show discussing how they are addressing each other; that dialogue can throw a wrench in the translation.

For Love is War, the source joke word was probably ちんちん (chinchin) which is both a childish way to say "penis" (like "poopoo" is a childish way to say 💩, or like "peepee") and a dog trick where the dog sits and begs (it might also have additional meanings). The translator probably had to think about how to make a similar gag with the existing visuals.

Translation is always a bit of a balancing act. If you translate literally you may lose the jokes, and to keep the same vibe you may need to go for something that's not a literal translation.

One example that comes to my mind is in FLCL there's a line in the dub about a store that still has Crystal Pepsi, however i the original Japanese it's a different discontinued soda. The translators made the call that almost no one outside of Japan would be familiar with the original soda, and that it wasn't really that brand itself that was funny, it was that this store was still selling a soda that was discontinued like a decade ago.

I feel like the weiner/hot dog/dachshund joke is probably pretty universal, it's a pretty simple sort of visual pun- dachshunds look like a hot dog (or maybe a sausage in countries that don't eat a lot of hot dogs) and hot dogs are pretty phallic, I suspect that pretty much every culture and language has similar kinds of jokes that translate pretty well. Maybe in some cases you swap out hot dog for kielbasa or salami or whatever the local tube-shaped meat product is and the accompanying local slang, but overall the joke isn't much deeper than "lol, penises are funny and these things look like a dick"

No idea with Japanese, but in Spain, most audiovisual media is dubbed.

Comedy is adapted according to context, both the plot's context and the current cultural context. Jokes are changed to fit the scene AND the translation.

When you start watching them in their original language, usually English, you find that the OG is usually better. And I say usually because in rare occasions, the joke in Spanish ends up being better, at least IMO.

The last name thing is true: outside family and close friends, that's what people call each other by.

I have no idea what the wiener thing was originally :)