Which tense can I use in these sentences and why?
Between 15.00 and 16.00 the day after tomorrow I have/am having/will be having/am going to have a game of tennis, so come to see me after that.
I’m studying hard today because I have/am having an exam tomorrow.
- Between 15.00 and 16.00 the day after tomorrow I have a game of tennis, so come see me after that.
Use have when you're talking about a schedule, a timetable (between 15.00 and 16.00 the day after tomorrow certainly reminds me of one).
- Between 15.00 and 16.00 the day after tomorrow I am
having† playing a game of tennis, so come see me after that.
I understand this to mean an arrangement has been made for you (normally by you) to play tennis at the specified time.
- Between 15.00 and 16.00 the day after tomorrow I am going to have a game of tennis, so come see me after that.
This sentence means you've decided and intend to have a game of tennis. It might imply you've previously arranged it, but the main meaning is that you're definitely intending to follow through with it.
- Between 15.00 and 16.00 the day after tomorrow I will have a game of tennis, so come see me after that.
This is merely stating a fact about the future. You're providing the listener/reader with a piece of information about the future.
Unless you have a reason not to use the first three constructions (i.e., what you want to emphasize isn't that something is part of a schedule, or something you intend to do) use will. Will is the default way of expressing the future.
- Between 15.00 and 16.00 the day after tomorrow I will be
having† playing a game of tennis, so come see me after that.
The construction above combines the meaning of will with the continuous aspect: you're stating a fact about the future, but you're adding that your game of tennis will be in progress at that time. This construction emphasizes duration; i.e., your busying yourself with something at some point in the future (this doesn't mean the action is longer). It conceptualizes the action as a progress, whereas will alone is more result-oriented.
Your second sentence is similar to the first one in that all the explanations above apply. The only difference is that you'd replace having an exam with taking an exam, for example.
† Have doesn't sound as natural as a non-stative verb in this sense, as explained by urnonav in their answer. In some other senses, however, it's possible to use it this way (e.g., We're having (= organizing) a party tomorrow).
All are, and away from any grammar jargon, correct. All are fairly straightforward in my opinion except maybe for am having, where we describe a future event as though we're living it at the time of speaking.. .
When you have something, it's a state. Actions can have continuous forms. A state is continuous by definition. Continuous forms of "to have", "to exist" or "to be" are odd because all of these verbs imply continued state anyway.
For both your sentences, I would suggest use of either simple present or simple future:
Between 15.00 and 16.00 the day after tomorrow, I have / will have a game of tennis, so come to see me after that.
I’m studying hard today because I have an exam tomorrow.
On the other hand, if you used more action-oriented verbs, continuous tense would actually be preferred:
I'm studying hard because I am / will be writing an exam tomorrow.
Between 15:00 and 16:00 tomorrow, I am / will be playing tennis, so come see me after that.
The day after tomorrow I _________ a game of tennis between 3 and 4, so come to see me after that.
This first sentence needs a bit of rearranging. Discussing such a small window of time so far into the future means that you're already talking about needing to do something on that particular day. Replying to their question, lead off with the day but still place the time after the event. For clarity's sake, you also want the time closer to your offer of availability afterwards.
Dealing with the verb 'have', Umonav's answer already covers the terminology pretty well. Something he didn't mention is that 'have' can be an action verb; it just almost exclusively concerns consuming food and beverages (I was having breakfast when suddenly..., e.g.) or planning (We'll be having the reunion over Memorial Day weekend, e.g.).
The standard way, which you mostly omitted, is to use will.
The day after tomorrow I will [be] play[ing] tennis between 3 and 4, so come and see me after that.
At that time in the future, I will do [or be in the middle of] this activity and you're not important enough for me to reschedule it instead of you. ['Go and ~', 'come and ~', &c. are more informal and technically correct formulations you sometimes see instead of 'go to ~', 'come to ~', &c.]
The day after tomorrow I shall [be] play[ing] tennis between 3 and 4, so come see me after that.
Some pedants like to retain the distinction between first-person 'shall' and second- and third-person 'will'. It's so uncommonly done these days that it probably even comes across as pompous even to the royal family. Unless your native language has trouble with terminal /l/ sounds (e.g. Mandarin-speakers) or your English teacher has a personal grudge against contractions, it's more informal and common to split the difference and use
The day after tomorrow I'll [be] play[ing] tennis between 3 and 4, so come see me after that.
which works for 'shall' or 'will'. ['Go ~', 'come ~', &c. are just asyndetic forms of 'go and ~', 'come ~', &c. and now more common than either other construction.]
The day after tomorrow I'm going to [be] play[ing] tennis between 3 and 4, so come see me after that.
This is more informal and variously emphasizes that informality; that the action is near term, as opposed to the immediacy of 'about to ~' or indefinite nature of 'will ~'; or that one is resolved that the action will occur in the manner 'shall' used to signify before everyone stopped using it in non-stilted speech.
The day after tomorrow I have a tennis match between 3 and 4, so come see me after that.
Pace userr..., this doesn't reflect a 'future of scheduling'. It's simply the present tense. I currently have this event scheduled, and you're not important enough to cancel or reschedule. ['Match' can imply a more serious contest than an informal 'game'; at the same time 'a noun of noun' is a more formal grammatical structure than 'attrib. noun'.]
