“A righteous man has regard for the life of his beast, but the tender mercies of the wicked are cruel.” Does it imply that the wicked are cruel to animals?
100% right! Cruelty to animals is a sign that a person is a rasha – wicked. This verse is cited often by the sages in reference to the various traits of the wicked.
When Jacob called together his twelve sons – representing the twelve tribes of Israel – to say what fate would befall them, Simeon and Levi were castigated and chastised for crippling oxen, among other things
Bereishit / Genesis 49:6-7
6 Let my soul come not into their council; O my glory, be not joined to their company. For in their anger they killed men, and in their willfulness they hamstrung oxen.
7 Cursed be their anger, for it is fierce, and their wrath, for it is cruel! I will divide them in Jacob and scatter them in Israel.
Are these verses also applicable for Noahides to exercise kindness to animals?
This translation is off. The Hebrew uses the word “shor,” ox, singular. Correct translation would be “In their wishes, they hamstrung an ox.” Rashi explains that the ox here is a reference to Joseph (See also Deut. 33:16-17). The word ikru, “hamstrung” in Hebrew, also means “uproot.” The commentaries explain that Jacob is making a play on words here, criticizing Shimon and Levi for trying to uproot Joseph.