Do animals feel emotions just like us or do they feel emotions on an
instinctual level only?
AN 3.61 says "for those who feel, I teach the Four Noble Truths". Animals do not "feel". If animals could "feel", they would not be controlled by their cravings & emotions. They would employ mindfulness & wisdom at sense contact & feeling. But instead of "feeling", animals are driven by their emotions rather than "feel" their "emotions". If animals could "feel" their emotions, they could practise vedanupassana & cittanupassana and stop the arising of craving.
What makes animals different from humans according to their planes of existence?
In Buddhism, the word translated as "animal" is "tiracchāna", which means to "move horizontally" or not evolve on a spiritual level; to merely be guided by instincts of reproduction, sex, territorialism, etc. For example, "worldly talk" by monks in Buddhism is called "animal talk".
The word translated as "human" is "manussa", which means "high minded". Suttas such as SN 56.47 define the "human state" as moral righteousness, non-harmful & realised the Four Noble Truths.
For example, animals believe pornography, sexual promiscuity & other forms of liberal sex are OK. Where as "humans" see the danger, degradation & dysfunction of sexual liberalism. The Lokapala Sutta says:
Bhikkhus, these two bright principles protect the world. What are the
two? Shame and fear of wrongdoing. If, bhikkhus, these two bright
principles did not protect the world, there would not be discerned
respect for mother or maternal aunt or maternal uncle's wife or a
teacher's wife or the wives of other honored persons, and the world
would have fallen into promiscuity, as with goats, sheep, chickens,
pigs, dogs and jackals. But as these two bright principles protect
the world, there is discerned respect for mother... and the wives of
other honored persons.
How can an animal get out of their plane? Is it by working off Kamma by suffering?
Animals leave the animal plane by realising the Four Noble Truths, as taught in SN 56.120, SN 56.47, etc, and abandoning instinctual lusts. This video (in English here) shows the animal named Angulimala realising the Four Noble Truths and having rebirth in the human realm due to abandoning violence.
How does evolution work in terms of animals/humans cause I hear it works with Buddhism?
Buddhism is about 'spiritual evolution' rather than 'physical evolution'. Sutta, such as MN 8, says wholesome states 'lead upwards'.
The Way Leading Upwards
Cunda, just as all unwholesome states lead downwards and all wholesome states lead upwards, so too:
(1) A person given to cruelty has non-cruelty to lead him upwards.
(2) One given to killing living beings has abstention from killing living beings to lead him upwards.
(3–43) One given to…to lead him upwards.
(44) One given to adhere to his own views, who holds on to them tenaciously and relinquishes them with difficulty, has non-adherence
to his own views, not holding on to them tenaciously and relinquishing
them easily, to lead him upwards.