Your use of them is correct.
Local dialects aside, "next Friday" is prospective (looking forward). It's what I would use right now to talk about the Friday a few days from now. By contrast, "the next Friday" is retrospective (looking back). You would use it in past tense, to refer to a Friday that is already in the past.
A clearer example:
Paul said, "I can't come to the party, I have a test next Friday."
Several months later, Paul heard Alice and Bob talking about how fun
the party was and sadly said, "I couldn't go to the party, I had a
test the next Friday."
Referring to the Friday after next Friday is largely a matter of dialect.
"The Friday after next" is the most common, and is prospective (like "next Friday"). The retrospective variation is technically "the Friday after the next", but I wouldn't use that; it sounds very stilted.
An alternative is "Friday week".
They do mean essentially the same thing, the up and off are mostly added for emphasis
Finishing off or up is generally used near the end of a fairly long task, or when specifically talking about the last portion of a task.
eg if I've got nearly a whole drink left, a friend is most likely to tell me
We'll finish our drinks, Jon, and get going
Whereas if I've got a couple of mouthfuls left he may say.
Finish your drink off, Jon, we're all waiting for you.
The latter both emphasizes the sentence (because my friends are waiting for me) and references the fact I'm nearly finished. The only real difference they apply is the emphasis - usually (but not always) requesting that the end of the task is expedited, or hinting that the task is taking too long.
They both tend to be added to long tasks, or tasks which are taking longer than normal, but this is by no means a rule. Similarly you'd tend not to add either to a short or quick task, as there's no need to emphasize... again, this isn't a rule, just a description of the most common usages.
Overall, off or up are virtually never necessary, they're colloquialisms added to informal speech. That said, they've made their way into formal speech too, and are fairly common.
If in doubt, you'll almost always be safe using "finish", but you'll also almost never be wrong adding "up" or "off" to the end.
(Note: the one caveat to this is that to "finish" someone would generally imply either killing them or knocking them out, if talking about a violent situation. "Finishing someone off", however, may imply something of an entirely different physical nature which I won't describe on StackExchange, but I'm sure you can guess...)
Best Answer
A noun phrase (NP) can be definite or indefinite:
A speaker marks an NP as definite when they assume the listener will be able to identify what the NP refers to. They mark an NP as indefinite when they don't make this assumption.
Some other words have semantics that are only compatible with either definiteness or indefiniteness. For example, same is definite; if you use it, you're specifying what the NP it modifies is referring to. There are two possibilities:
Since same only make sense in definite contexts, it can combine with the, and it can't combine with a. For this reason, *a same name is not standard English.