When you want to say that you have to have something or that you need to have something that POINTS TO A CAUSE (ex: you need something to complete a task), you would use: -なければいけません/ないといけない Rather than saying, "I need to have something," in Japanese, we say, "If I don't have something, then it (the situation) can't be," or, "If I don't have something, then I can't go on [to complete this task]." And remember contractions for ease of pronunciation: (Formal) なければいけません --> ないといけません //// (Informal) なければいけない --> ないといけない So, let's say you want to bake a cake, but you don't have eggs. You would say, "たまごがないといけない。” or "たまごがないとできない" or "たまごがないと作れない" (you get the point, right?) However, in your situation, since you want a cat JUST BECAUSE, saying, "ねこがないといけない" would sound a bit unnatural because it sounds like you're saying, "If I don't have a cat, then I can't do ____." So, instead, it would sound more natural to say, "ねこがほしい" (simply "I want a cat"). Or, if you want to be more dramatic, you can say, "ねこがないと生きていかれない" ("If I don't have a cat, I can't go on living.") Also... There is no kanji required for this since "なければいけません/ないといけない" is purely Japanese and has no Chinese influence. Hope this helped!

We need the object for "have" to explain it because "have" has several translations depends on the objects. However, generally, the translation for HAVE TO is 〜しなければならない （～shinakereba naranai) and the formal form is 〜 しなければなりません （～shinekereba narimasen).