Clauses, and comma before “nor”

Solution 1:

They can stand alone as sentences:

I do not like biology. Nor do I like chemistry.

Therefore they are independent clauses.

They can also work with a comma as you suggest, or a semicolon:

I do not like biology, nor do I like chemistry.

You could use a dependent clause to express the same thing:

I do not like biology nor like chemistry.

(Valid, but awkward in this particular case. A comma would be valid but not necessary here, and probably improve it)

Or just simple conjunction on the nouns:

I do not like biology or chemistry.

Solution 2:

They are two independent clauses. They could be expressed as

I do not like biology, and I do not like chemistry.

The word nor serves as both a conjunction and a negation, standing in for the and and the not.

Macmillan defines nor as a conjunction

used after a negative statement when adding another negative statement
I have not been asked to resign, nor do I intend to do so.
She could not speak, nor could she understand anything we said.

While in declaratory independent clauses, the most common word order is subject-verb, common usage reverses the standard order in the second independent clause to verb-subject when using nor.

Solution 3:

Yes, these are most definitely two independent clauses. It's like saying "I do not like biology, and I do not like chemistry." But in the cases of voicing a negative, we use "nor". This, however, is changing in many regions of the U.S. Some actually omit the "nor" and use "and". "Nor" is used mainly in writing and not so much in spoken American English. People do not prefer to use it, nor are they likely to change.