Are there /ɔ/ and /ʌ/ sounds in informal American English?

I read a book about American English. It reports that, in standard informal conversations, American English doesn't use the /ɔ/ sound; it uses the /ɑ/ sound and /ʌ/ and /ə/ are not different. Are they really?
That book would not use the /ɔ/ and /ʌ/ sounds, but when I look in my American English Dictionary for some words, such as more, door, and love, they are reported to be pronounced /mɔr/, /dɔr/, and /lʌv/.
How should I pronounce these words, if there are no /ɔ/and /ʌ/ sounds? Should they be /mɑr/, /dɑr/, and /ləv/?
Can /ɑ/ sound replace /ɔ/, and /ə/ replace /ʌ/ in every word?
What about formal American English? Does it have /ɔ /and /ʌ/ sounds or not?

Solution 1:

The /ɔ/ as described by Wikipedia is an unelisioned "aw" sound, as in the pure Latin vowel "o" for those singers out there. The /ʌ/ sound, as found in the word "plus", is an open-backed "uh". Close off the back of the throat by dropping the soft palate, bringing the jaw back and/or raising the back of the tongue and you have what most Americans would call a "schwa" (/ə/) as in the second syllable of "special".

In most American English dialects/accents, the mouth is held in a more open, relaxed position while speaking than for most British accents. This can tend to make "aw" sound more like more like "ah", and "ah" like an open short "a" as in "bat". Similarly, the opening of the lips leads to closing off the back of the throat to provide nuances between vowels, which can make "uh" sound like "ugh", "oo" sound like "eu", etc. Vocal coaches tear their hair out over this natural accent, especially in the deep South and Texas, where the "twang" pollutes the "pure" Latin vowels normally desired for singing in almost any language.

However, I doubt you will find an English dialect where a particular vowel shape is never heard. First of all, there will always be a word in the language that even a heavily-accented speaker will pronounce using the shape you're looking for. It may be misplaced, but it'll be there.

Second, there are degrees of vowel modification from what we would consider "neutral" American English to "accented", no matter the accent, so you will always find a person who speaks with just the right level of accent to use the vowel shape you're looking for at least some of the time. Urban residents tend to accent less than suburban and rural in my experience, and higher education, which normally involves a mingling of people from many locations nationwide, also tends to reduce accenting. Watching national broadcast TV also tends to reduce accenting, as most actors have their natural accents trained, coached, and/or beaten out of them while on camera.

Solution 2:

Standard English has all of the sounds you mention, but, yes there are some quirks.

Some dialects of English don't distinguish between /ɔ/ and /ɑ/; this is known as the caught-cot merger. It is so called because caught and cot are both pronounced the same: (/kɒːt/ or /kɑt/ depending on the region). As you can see in the Wikipedia article and the accompanying map, some dialects have merged these vowels together, but many have not.

Now, as for /ə/ and /ʌ/ — AmE does have both of these sounds, but in most cases the pattern is totally predictable. In stressed syllables, /ʌ/ can occur, while in unstressed syllables, only /ə/ is used.

In Standard British English, there is more use of /ə/, in part because Standard BrE doesn't pronounce /ɹ/ (henceforth /r/) syllable-finally. So a word like nurse, which in American English would be pronounced /nɝs/ (with an r-colored vowel), can be pronounced /nə:s/ in British English (though it isn't always).

So, with this information in mind, on to your examples:

"More" and "door"

Examples with syllable-final /r/ are generally going to be special, as indeed these are. In Standard AmE, these are pronounced /mɔr/ and /dɔr/, as you read. In dialects that don't pronounce /ɔ/, the words are pronounced /mor/ and /dor/. In situations that aren't r-colored, it is indeed /ɑ/ (again, in certain US dialects).

"Love"

Following the rules I laid out above, you may have figured out that love is pronounced /lʌv/ in AmE, as it is a stressed syllable. But, if it were unstressed, it would theoretically be pronounced with a schwa.

So, let's take a different example: the word "just" /dʒʌst/ is sometimes stressed and sometimes unstressed. When unstressed (often when saying something like "just do it already"), the word becomes /dʒəst/.