I would rather say 'no'. It is true that we usually write animals' and plants' names in カタカナin biological field. So I guess you saw biological categories, such as 'order' and 'family'. They are always written in カタカナ. In additions, you can also find this trend in media. However, we also use ひらがな and 漢字 in literature, essay, SNS and so on. I have no idea of distinction, but there are tendencies about some of those words. 漢字：犬、猫、魚、鳥、馬、牛、桜、梅、菊... ひらがな：うさぎ、きつね、たぬき、にわとり、ねずみ、ひまわり、あじさい... カタカナ：カラス、カエル、ハト、アヒル、ホウセンカ...

Katakana-notation emphasize that it is a name of species. Wikipedia, like all the other encyclopedia, describes an animal itself as species. For example, there are several way to write Hito in Japanese. 人:a person, in general ひと:a person, not usual, perhaps a special one like 好きなひと ヒト: Homo Sapience. Maybe in contrast to other animals. The same applies to other animals. Hiragana gives a friendly impression. Katakana looks objective and distanced. Kanji is rather neutral.

from wikipedia on katakana: Technical and scientific terms, such as the names of animal and plant species and minerals, are also commonly written in katakana.[5] Homo sapiens (ホモ・サピエンス Homo sapiensu?), as a species, is written ヒト (hito), rather than its kanji 人. Katakana are also often, but not always, used for transcription of Japanese company names. For example, Suzuki is written スズキ, and Toyota is written トヨタ.

In my experience, no.

Japanese people tend to use katakana with animals and fruit. Nowadays they use katakana with other words because it seems to be cool I guess? haha. There's no rule for that. It just is the way it is.