Yes, German tends to have more exceptions to its rules than instances where the rules apply. That's a classic complaint -- Mark Twain wrote about it in "The Awful German Language".
Note that the plural of "Apfel" is "Äpfel", so it does change. There are a few words where a vowel changes to an umlaut in the plural: "Baum" -> "Bäume", "Ofen" -> "Öfen". An example where the plural is the same as the singular is "Teller".
Das Ei is a neuter noun. Plural -e is used with masculine and feminine nouns.
A question re. pronunciation: the "r" ending eier is not pronounced here. Is that hoch deutsch, plat deutsch, or another dialect? A relative of mine pronounces "r" almost gutterally, similar to French: "garten" sounds like "garchten". Perhaps a native German could explain. Thanx in advance!
My guess is that your relative is Rhenanian or Hessian. For these dialects, a hard pronunciation of the r before a hard consonant is characteristic. In other dialects -- that of the Ruhrgebiet for example -- the r is slurred, so it sounds more like "Gahten". Others might use a rhotic r. But all dialects I am aware of drop the 'r' at the end of "Eier" -- if you pronounce that 'r' it's going to sound like a French or Slavic accent.