The first sentence is fine, except for a small error of idiom—you need to say "five years old", or use some other expression such as "would have celebrated its fifth anniversary" or "would have lasted for five years".
The second sentence is formally incorrect, but will pass in ordinary conversation as equivalent to the first.
The third sentence means something different.
Your confusion probably arises from two considerations:
- You are describing the conditional relationship between a past event, the ending
of the relationship, and a future event, the anniversary.
- Preterite would has a double function: a) as the past form of will, and b) as the "unreal" form of will in a conditional.
(In what follows I am going to use non-past/preterite for the forms will/would and are/were, and I will reserve present and past for the time references.)
Perhaps a look at some alternatives will clarify what's going on here grammatically. First, A, we'll look at the universe in which Steve and Anne have not yet separated; then, B, the universe in which Steve and Anne have separated; and finally, C, we'll look at how this applies to your sentences.
A. Steve and Anne have not separated.
If you are speaking about a present relationship, you use the plain non-past forms for both the condition (IF) and consequence (THEN) clauses.
If they are still together their relationship will turn five years old next month.
If you are speaking about the same situation as a past relationship—for instance, if you are Anne's biographer, writing in 2040—you would write the same sentence with the two verbs are and will "backshifted" into the preterite forms were and would:
In 2013 Anne realized that if they were still together their relationship would turn five years old the next month.
(Note that you also have to change "next" month to "the next" month.)
B. Steve and Anne have separated.
If you refer to this as a present condition, you use the preterite forms in both the IF and THEN clauses to mark these as "unreal":
If they were still together their relationship would turn five years old next month.
This sentence describes an "alternate timeline" which begins in the present, with Anne and Steve together right now.
Note that this sentence is almost identical to the one in A2—the only difference is that the next has reverted to plain next. This may seem confusing, but in practise it almost never is, because you and the people you are talking/writing to know whether you are speaking about the present or the past. It's only when sentences like this show up out of context, as in a textbook, that confusion arises.
But what happens if, as biographer, you want to describe this as a past event? You've already "used up" all your available "preterites* to signify unreality; how then do you mark pastness?
Our solution for this is to use the "perfect" construction, with HAVE + past participle; the tensed verbs which head the constructions are cast in preterite form, had and would, to express the unreality, and the "perfect" constructions express the pastness.
If they still had been together in July of 2013 their relationship would have turned five years old the next month.
This sentence describes an "alternate timeline" which begins in the past, with Anne and Steve together then.
C. Your sentences
Steve and Anne have ended their relationship, which would have turned five years old next month.
You write this in the present—the main verb is cast in the "present perfect", have ended.
However, the event which it describes, the ending of the relationship, took place in the past.
And your second clause is the consequence (THEN) clause of an unreal condition (IF) clause. You don't make that unreal IF clause explicit, but it has to be based on what you have already written, on the past event described in your first clause. Consequently, it is written as if the "alternate timeline* began in the past, as in B2, with Steve and Anne not ending their relationship:
Steve and Anne have ended their relationship, which [if they had not done so] would have turned five years old next month.
In conversation, which is often imprecise, you may employ the bare preterite would here; but in formal use you must give your conditional clause an explicit present reference:
Steve and Anne have ended their relationship, which would turn five years old next month if they were still together.
As to your third sentence: First, be supposed to today usually implies moral or practical obligation, equivalent to ought to. Second, even if you use suppose in the old sense of "expected", you are no longer talking about logical consequence, as in your first two sentences, but about what is believed. There is no conditional construction implied here; it deals with a different sort of discussion.