This is a really broad question, and I don't want this answer to get too long. If my answer to one of your questions is too brief, I suggest asking another question with more specifics. Posts like this typically should be asked as three separate questions, but I'll do my best.
In the sentence 明日、テストがありますね what is the point of having が instead of は?
This is a good question. This highlights one of the key differences between は and が.
With は, you have the topic marker. When you say テストはある, you technically say that there is a test, but that's no different from normal. The impression is that tests happen all the time, so tomorrow isn't any different. Use of は here can come off as nonchalant, casual, or even cocky.
With が, you get a subject marker. When you say テストがある, you say that the test is tomorrow. There may or may not be tests on other days, but the test is tomorrow. I is it inherently more specific and target to a specific subject.
Here's a great post on the difference between は and が. I highly suggest you read it, as the differences are very subtle to the new Japanese speaker.
What's the difference between wa (は) and ga (が)?
In the sentence このへんに本屋がありますか how is に being used here? I learned that に is only used for movements like いきます.
As you have no doubt seen in the comments, に is not limited to indicating a direction of travel. There are uses in passive and causative structures to indicate actors. It can be a particle indicating location of existence, it can indicate the result of a change, it can be an object of a verb, it can be the source of a verb, it can indicate a specific time, and it can even be the Japanese equivalent of the English per. All of these uses of に can be found on this web page. Once again, I suggest you study this source.
In this particular case, に represents a location of existence. I would translate the sentence as:
Is there a bookstore in this area?
(linear translation as follows:) This area (このへん) in (に) bookstore (本屋) exist (ある)?
While in this case, に can translate to 'in,' do not consider that a 1 to 1 translation. There area cases where the location of existence particle does not translate to 'in.'
For this sentence, the speaker previously received the listener's phone number after asking them. The speaker then thanks the listener in the sentence ありがとう。じゃ、今度、電話をしますね。What is the point of ね here?
The simple answer is that in this case, ね provides emphasis to what is being said. In this case, I would say that the English equivalent is:
I will call you tomorrow.
What's really happening is that the speaker is indicating a decision after some consideration.
ね, has other uses however. (I used this source heavily for the following examples.
I recommend you read it.)
You can use it like to ask for agreement:
むずかしいね。(difficult, isn't it?) In response, you can agree with ね as well:
You can use it to confirm information:
がっこうにくるね。 (you're coming to school, right?)
You can use it to express something after consideration (as is the case with your sentence).
You can use it to express denial:
げんきじゃないね。 (No, they aren't lively.)
You can use it as a filler word:
あしたね、いくつもり？ (Are you going tomorrow?) In this case, ね doesn't really have meaning.