"As well as" means "in addition to"--it does not mean "and."
You would say
The teacher and the students are busy
but phrases such as together with, as well as, and along with do not function like and, that is, they do not create compound subjects.
And, you should include commas:
The teacher, as well as the students, is busy
This is because "as" (as well as), "to" (in addition to), and "with" (together with, along with) are prepositions and what follows them is not a subject, but rather an object (of the preposition). Hence, anything following these phrases does not affect the conjugation of the verb. Many people find this difficult to digest, and often it is better to recast the statement--by replacing "as well as" with "and," for example, or saying
The teacher is busy, and so are the students
or some such.
As for "Horse is a faithful animal," that is an entirely different question, and deadrat explains nicely above.