∗ I'll be at my uncle's house you should need to reach me
is ungrammatical. You need an if in there: if you should need to reach me. The if is only omitted if should is inverted with the subject. The inverted construction is a ‘fossil’ from much older usage; today it is fading into disuse, and has a very quaint ring in the US. Once any verb might be used in this construction, but today only should, had and were are used this way.
The rest of your sentences are all ‘relevance’ conditionals: that is, the protasis (‘condition clause’, ‘if clause’) does not express a condition under which the apodosis (‘consequence clause’, ‘then clause’) is true but a condition under which the information in the apodosis is relevant. For the most part they all express the same thing, which may also be expressed with an ordinary if clause:
I’ll be at my uncle’s house if you (should) need to reach me.
However, there is another dimension to the versions with (just) in case. In case may express, alternatively or additionally, the reason why you will be at your uncle’s house: you will be remaining at a particular place (rather than going out) so that you can be reached if that becomes necessary. That is the primary meaning of the expression, as in
I’ll take an umbrella in case it rains.
A conditional sentence has two parts: the "if-part", or protasis, and "then-part", or apodosis.
In the apodosis, we use would to create that "conditional" feeling. So the sentence
If we had had lots of money, we should have travelled round the world.
would be illogical: the protasis clearly says that we did not have money during some period in the past, but the second half of the sentence is not an apodosis, because it uses the wrong modal verb. The second half looks like a normal sentence on its own:
We should have traveled around the world! (instead of doing some other things in the past, we should have traveled around the world)
This clause implies that we had the ability to travel around the world. It contradicts the protasis.
The same applies to your conditional 2 example:
If I worked harder, I should pass the exam. [improbable situation in the present or future],[moral obligation]
The first part invites some apodosis with would. The second part just plainly states your obligation to pass the exam. But according to Wikipedia,
Occasionally, with a first person subject, the auxiliary would is replaced by should (similarly to the way will is replaced by shall). (Wikipedia says this about the apodosis of both the second and the third conditional)
So maybe we can use should there after all, since I is a first-person subject. Let a native speaker decide.
The use of should is usually "deontic" (what should be: used to express norms, expectations, speaker's desire) while the use of would is "epistemic" (what may be).
The modal verb should could be sometimes used in the epistemic mood too, so we can come up with sentences such as
If I worked harder, that should be surprising!
But the effect would be comical and ironical, not the usual effect with the conditional sentences. The two halves of the sentence would still look somewhat disjointed.
Best Answer
Yes, should and the other verbs are used properly in your sentence. With StoneyB's suggestions, the rest of the sentence is okay, too:
Of course, this isn't really a proofreading site, and StoneyB covered the basics about how the verbs fit together, so I'd like to focus on the specific question you asked about should. Here are some details about how should works in this sort of sentence:
Conditional should
Conditional should appears in conditional sentences with if, adding a small amount of modal meaning, specifically a low degree of doubt:
There isn't much difference in meaning between these two sentences, because should doesn't add very much modal meaning to the sentence. But because it does add some meaning, if there's no doubt at all that the conditional is true, should cannot be used. Imagine a boy criticizing his own father for not helping him:
Example 2b is marked with a # symbol to indicate that although it's grammatical, it doesn't make sense in the imagined context. He knows that the listener is father, so there is no doubt whatsoever, so it doesn't make sense to use should to express doubt.
Note that in example 2a, the verb be is in its finite form are, contracted to 're. In example 2b, we added the modal auxiliary should, so the verb be is now in its plain (infinitive) form. We'll see the same thing in your sentence below.
Inverted conditionals
Besides adding should, there's one more element at play. Certain conditional sentences can be rewritten with subject-auxiliary inversion, omitting if:
These sentences convey the same meaning, but grammatically there are two differences. First, we've swapped the subject I with the auxiliary had. Second, we've removed the conditional if from the beginning of the sentence. This style is markedly formal and somewhat less common than the more basic style.
Conditionals with should are often written in this style. Let's start with a basic conditional:
Now let's add conditional should:
Last, let's place it in the formal inverted style. We'll have to invert the auxiliary should with the entire subject noun phrase the construction of the University Hospital, and we'll be getting rid of if, too:
And now we've got your sentence, similar in meaning to 4a, but more formal in tone.
The explanation above is based on information and includes examples from Huddleston & Pullum's Cambridge Grammar of the English Language (2002), and can be found on pages 188 (where conditional should is described) and 753-4 (where inverted conditionals are described).
In this answer, the # symbol indicates that the sentence doesn't make sense in the intended context.