[W]hen they say a long vowel, are they deliberately saying one long vowel sound or two of them directly following each other?
If this is about phonology, as the tag indicates, the answer will be: two, or neither (at least in Standard Japanese).
It's merely two same vowels adjacent by chance when in between two words, or between word stems and inflections. In your examples, 唐揚げ apparently consists of two words (morphemes) kara + age, thus the two cannot be merged into one in a speaker's mind. いい, although being short, is made of i (stem < yo) + i (adj. ending) and follows it too.
Meanwhile, it's a short vowel followed by a lengthener, when seen inside an indivisible word. 教室 and 先生 in your examples fall under this (have nothing to do with orthographical spellings, just to be sure). By "lengthener" I mean, we recognize a prolonging phoneme //ʀ// as one of moraic phonemes of Japanese (the rest is, moraic nasal //ɴ// a.k.a. ん and geminator //ꞯ//* a.k.a っ, as far as widely accepted among researchers). //ʀ// has no sound value by itself, but can make previous vowel a mora longer. Phonemic representations of 教室 and 先生 are respectively //kyoʀsitu// and //seɴseʀ//.
So what makes you happier if we assume a //ʀ// instead of a long vowel? I think this page I found makes a fairly neat summary on the advantages of //ʀ// analysis. It says (with adapted terms):
- When you transpose morae in wordplay, 貧乏【びんぼう】 "poor" with first and third mora interchanged should be ボンビー, which is only accountable if the word was //biɴboʀ//; if it were //biɴboo//, the result would be //boɴbio// ボンビオ.
- A series in demonstrative paradigm こう ("in this way"), そう ("in your/its way"), ああ ("in that way"), どう ("how") would be more consistent represented in //koʀ//, //soʀ//, //aʀ//, //doʀ//, rather than simply a long version of each vowel.
- (omitted; it's about distinguishing two short vowels and one long vowel.)
cf. the Wiktionary entry of 里親 ("foster parent"):
Compound of sato (“village”) and oya (“parent”). Often cited in contrast to satōya (“sugar dealer”) in discussion of the phonological distinction between long vowels and geminate vowels.
*It's a small capital Q, which has only recently been included in Unicode.