Your grades have gotten steadily worse since the beginning of the term.
In this sentence, I understand that the grades do not keep being worse as time goes on because of “gotten”, that is, the degree of worse grades is likely to have stabilized at some point.
But do the grades keep being worse as time goes on because of “steadily”?
Let me say another examples.
“I have become thinner since I have gotten older.”
“I have become thinner since I got older.”
What is the difference between two sentences.
(Or between “since the term began” and “since the term has begun”)
Maybe the examples are not suitable.
But I want to know the difference between “since + past verb” and “since + present perfect”.
To answer your question on the difference between the two verb tenses you need to recognize that the simple past is used to express the idea that the action is over and done with at some point in the past. The present perfect is used to express that the action started sometime in the past and continues on in the present or just very recently ended. Here is an example:
I ate my dinner. (and now I am done with it.)
I have eaten my dinner. (and I just finished it.)
I have eaten dinner at 6:00 every night. (I did in the past and will continue to do so in the future.)
In your first example sentence about the grades, the grades "have gotten steadily worse" because you have used the present perfect tense which indicates that the action is still happening. The adverb "steadily" is indicating that the grades are falling at a consistent/regular rate, but is not affecting the verb tense you used.
Your example sentences all use the word "since" in them and I think that may be part of your confusion. The word "since" is an adverb that modifies the verb and tells when the action is happening. But again, "got older" is over and done, and "have gotten older" is a continuing process. Frankly, you are right that this example probably isn't the most clear in helping understand the difference in meaning between the tense because you can't be over and done with growing older or you would be dead.
In your last examples, "since the term began" means that the term began at some point in the past. "Since the term has begun" means that the term began at some point in the recent past.
Hope all that helps clarify the two verb tenses!