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/.

Are there /ɔ/ and /ʌ/ sounds in informal American English?
I am using the client object model and I want to access a List using the server relative Url.
Folders can be accessed using the web.GetFolderByServerRelativeUrl but no such function exists for lists.

The available API is web.Lists.GetByID and web.Lists.GetByTitle which I can't use (The list title is different from the listname/listpath)
