The term "sentence" is actually very vague. It is a syntactically dependent unit, which has at least one finite verb and a subject (which may be implied). What we usually mean by a sentence is a unit that begins with a capital letter and ends with a full stop (period, if you're American), or an exclamation mark or a question mark. A unit that includes a subject and a verb phrase can be usefully called a clause.
Thus, if we say "Willie sees the bird; Frank hears the noise", we have one sentence with two clauses.
The semicolon is not a replacement for the colon. Semicolons are used primarily to separate what are called contact clauses.
We can combine clauses into longer sentences in a number of ways. We can have relative clauses:
The woman who is wearing the yellow hat is the Duchess.
We can have subordinate clauses:
If I had a choice, I wouldn't be here.
And we can have coordinate clauses:
Willie sees the bird and Frank hears the noise.
Coordinate clauses are joined together into longer sentences using a coordinating conjuction: usually "and", "or" or "but". If we omit this conjunction, we have a sentence with two contact clauses -- and that's when we need the semicolon.
In your second example, you have a conjunction, so you shouldn't use the semicolon: "I felt sick, therefore I stayed home". Again, you can omit the conjunction, and then you get contact clauses: "I felt sick; I stayed at home."
Using commas instead of semicolons is frowned upon; you get what are called "run-on sentences", and they are considered inelegant and, by some, illiterate.