Generally 種違い or 腹違い is not considered as rude/discriminative, it is simply a bit oldish. In not-so-frequent occasions to refer to such things, I guess 父親/母親が違う would be used.
That said, I won't be surprised if there are people who claim they should be avoided mainly because 腹 or 種 may sound treating people as an instrument to produce children.
As for ハーフ, from the Japanese side, it has no derogatory sense and means people whose parents are of different nationalities, although it is usually expected that there's a difference in race. (E.g., a child from a Japanese of Japanese origin and an American of Japanese origin would be ハーフ technically, but not typically thought of when speaking of a ハーフ.)
Not sure how it sounds to ハーフ people who understand English natively.
This Wikipedia article contains more details on ハーフ, but for most ordinary people, it does not sound rude. It mentions there is a word ダブル to avoid ハーフ, but I've never heard of it (sounds like coffee or whisky...)
The word あいのこ is derogatory, but this is rather dated and not really current.