Can present tense be used without an adverb like "usually" or "always" when describing a habit?

My dog always sleeps under the bed

and

My dog sleeps under the bed

both convey the notion in both being that your dog habitually sleeps under the bed.
If you intend to completely precise, and your dog occasionally sleeps somewhere other than under the bed, then

My dog sleeps under the bed

will be more appropriate.

It usually sits on the sofa

is fine. There is an excellent chance the dog sits somewhere other than on the sofa.

It sits on the sofa

is also OK As far a grammar goes, both example sentences are fine. The adverbs would serve to create a fine, exact meaning. If that is desired, the adverbs should be used.

Yes. Complete, grammatically correct sentence. Basic SVO structure with dependent clauses (led by prepositions "under" and "on"). Can come across as awkward in some contexts, but is actually a common construction. Listen carefully in conversation and you'll hear it.

Yes. Not only is it OK to leave off the unnecessary adverbs, I find it preferable to avoid useless words. The exception would be when the adverb is important. For example, if you are going to continue the sentence about your dog usually sleeping under the bed with a story about a time he didn't, then using the word "usually" would play an important role in introducing the following sentences.