Translation:I have a big brother and a little sister.
You'd say Ich habe einen kleinen Bruder und eine kleine Schwester.
In the plural, a single adjective works: Ich habe kleine Brüder und Schwestern.
I think the "problem" might be the fact that countable nouns need a determiner in the singular -- something like Ich habe einen Bruder und Schwager for "I have a brother and brother-in-law" sounds wrong to me in both languages (unless you are saying that your brother is simultaneously your brother-in-law) and it would have to be einen Bruder und einen Schwager "a brother and a brother-in-law" with an einen / "a" in front of both nouns.
With Bruder und Schwester, you run into the additional issue that they are of different genders and so they can't share an article anyway.
With uncountable nouns of the same gender, you could probably get away with one adjective for both: Ich habe alten Wein und Champagner gekauft "I bought old wine and champagne" (though as in English, it would be ambiguous whether the champagne was old or not).
With different gender, you have adjective ending troubles again: Ich habe alten Wein und Bier gekauft can only mean "old wine, and also beer" -- the beer cannot be old because that would be altes Bier and not *alten Bier.
You use nominative for the person or thing that is ‘doing’ the verb, so to speak. You use accusative for the thing or person which the verb is being done to. In English grammar, these are the subject and the direct object respectively. In German grammar, we use nominative and accusative cases instead.
So in this sentence, ‘ich’ is in the nominative, because (using English) it is ‘I’ who has the brother and sister - I am doing the having. The brother and sister are in accusative because, in a sense, the act of ‘having’ is being done to them. This changes the articles (i.e. forms of ‘the’ and ‘a’) from their nominative forms, ein and eine, to their accusative forms, einen and eine. The feminine (and neuter and plural) form of the article happens to be the same in nominative and accusative, but don't let that stop you from realising it is in a different case. You only have to worry about using the different form of the masculine one for these two cases though.
So, to demonstrate using the verb ‘sehen’, meaning ‘to see’. (using ‘mein’, which takes the same endings as ‘ein’)
Ich sehe meinen Bruder. - I see my brother.
Mein Bruder sieht mich. - My brother sees me.
In the first, ‘meinen Bruder’ is accusative because he is the one being seen, whereas in the latter, ‘mein Bruder’ is nominative as he is doing the seeing.
With "big", it's a piece of cake, because "bigger" has an umlaut (größer) and "big" does not (groß).
With other nouns, you have to see whether it's before a noun or not -- for example, kleiner could be a form of klein (ein kleiner Kuchen "a small cake") or a form of kleiner (dieser Kuchen ist kleiner "this cake is smaller").
If you wanted to use "smaller" before a noun, then you have, for example, ein kleinerer Kuchen "a smaller cake", with one -er for the comparative and another -er to agree with the noun.
Because it is not the most direct translation of the German sentence. Sometimes compromises have to be made about how far removed from the literal translation an accepted translation can be if it still means the same. I think in this case "older" and "younger" are better not accepted because they are comparative forms, while the original German adjectives are in the base form. In a lesson about comparative adjectives, you can see how that could confuse a learner.
Adjectives before a noun need to have an ending.
The ending depends on the gender, number, and case of the noun as well as on what comes before the adjective.
Here, you have einen before the adjective so you need mixed inflection, Bruder is masculine singular and - in this sentence - in the accusative case. So you need an ending of -en.
Schwester is also after the indefinite article and is also singular and accusative, but feminine, so it takes the ending -e.
Sentences don't have cases.
Parts of sentences have cases -- to show their role in the sentence (e.g. as subject or object of the main verb in a clause, or depending on a preposition if they are in a prepositional phrase).
In this sentence, ich is in the nominative case, einen großen Bruder is in the accusative case and eine kleine Schwester is also in the accusative case.
Schwesterlein is a word, but it's most often used when speaking to your sister -- it means something like "sister dear" in practice: the -lein doesn't necessarily mean that she is smaller or younger than you but makes the word "cuter".
If you talked about your sister and said mein Schwesterlein, it would sound (cute and) odd.