In English, the sounds like a, e, i, o or u are called vowel sounds.
I want to know, whether ä, ö, and ü have certain names in German language and what the dots used to produce them are called.
Multiple questions, multiple answers:
In English, the sounds like a, e, i, o, u are called vowel sounds.
Erm, technically true, but you’re forgetting y. And there are a lot more vowel sounds that just the five ‘standard’ ones. There are some twenty different vowel sounds in English if you include diphthongs (and there’s no real reason not to).
I want to know, do these sounds like ä, ö, ü have certain names in German language?
Germans would consider them vowels just like the other five (y is a variant of either ü or i) and the diphthongs (ei, au, eu, ui) German language has. If there is, for any reason, a need to specify that you’re talking about and only about the three dotted ones, the terminology is Umlaute. If you’re just talking about a specific one, it’s Ä, Ö or Ü, just like they’re pronounced.
what [are] the dots used to produce them […] called?
A.k.a. the question in the title. These dots, usually called trema (or diaeresis) in English, are called Umlaut in German (yes, this is the same word as above for the vowels themselves) if and only if they are used to signify Umlaute.
If you’re talking about a word like naïve, or the Albanian letter ë, the only correct terminology is I/E mit Trema. This also occurs in generic German names (Hoëcker) and old-fashioned spelling (Alzeÿ). Note that German generally prefers the word Trema where Umlaut is not applicable, while English prefers diaeresis for the naïve-case and trema only for distinct non-umlaut letters. (In the Alzeÿ-case, Diärese would be outright wrong.)
The word Trema can also be used in the case of umlauts. So it’s perfectly acceptable to say A mit Trema.
Colloquially, the dots also have a variety of other names derived from their shape: Punkte, Doppelpunkt, Pünktchen, Striche, Stricherl, Strichelchen and more. Note again, that any Strich-type variant is only applicable, if you’re talking about ä/ö/ü. Ï and ë would only get names that derive from Punkt in some way.
In case you’re interested: Trema originates from Ancient Greek and is a name based solely on the shape of the diacritic.
Umlaut, as you might have guessed, stems from the German word which means changed sound.
Diaeresis is also Greek, meaning division or separation. It is only truly appropriate in cases where the trema shows that two letters are to be pronounced distinctly.
 This is due to the origin of the tremata. For the three common German letters, they derived from a superscripted e, which in blackletter and even more in Kurrent handwriting resembles a pair of vertical lines. Eventually, the e got reduced to the vertical lines which got further reduced in typesetting to a pair of dots. For diaeresis cases, the dots did not derive from lines but were introduced as dots originally.
The result of the dots, so the letter with the dots on it, is an Umlaut – literally a “resounding” – of the vowel.
The dots themselves are commonly known as ä/ö/ü-Striche (or Strichelchen), depending on what word you have in mind. That’s what we, including the teachers, used in school. The term umlaut only came to my attention once I started having contact with English speakers who would learn German. But that might just be me. The term trema however is widely … unknown. I'd recommend not using it.
To be absolutely accurate, “Umlaut” is a technical term for the phonological/historical process which transforms - for example - /u/ to /y/. This is the only meaning recognised in the standard dictionaries (DWB, DWDS, Duden). The two dots on top of some letters are properly called “Pünktchen”.
Diakritische Zeichen is a term covering trema, any kind of accent, cedille and other letter "ornamentation" without the letter itself.