When Beijingers use words like 一点儿, the ending syllable will be very pronounced as far as I understand, and I'm aware that this is a typical feature of the Beijing dialect. But how is it pronounced in the rest of China? Is there no "er" sound there at all, or is it only there for special cases "e.g. 一点儿"?
In MOST cases... Outside of Beijing, in texts, I believe the 儿 is still pronounced. But you can be sure that in spoken Chinese, it will never be pronounced (unless some kids are trying to mock the access by over accentuating it).
I said in MOST cases because there are some words that have simply been adopted by non-Beijing'ers and will always be pronounced with a 儿, and would sound weird otherwise.
So whether you're from Beijing or not, you will always pronounce and write this word as
The closest I can come to describing this in American English is how we pronounce certain areas/things in the actual dialectical origin of that word. For example, we say "New Orleans" as someone from New Orleans would say it: "Norlins" even though we're not from New Orleans necessarily. If you say "New Orleans" and pronounce every syllable, it would sound weird to anyone... native or not to New Orleans.
EDIT: a better example is Louisville Kentucky. EVERYONE pronounces Louisville as "Louville".
Across the Northern China, 儿 is usually realised as an /r/ sound gluing to the previous sound (and may affect the previous sound in some accents). It is not pronounced independently as a character.
In southern China, -儿 construction (or the so-called 儿化) is much less common, which only exists in a few phrases such as 一会儿 (actually I cannot think of a second example at the point), and 儿 is usually realised as a separate character. People tend to say 一点 instead of 一点儿 in southern China. (This could be used as a linguistic marker to tell if one is from northern or southern China.)
Answer to each question
Pick one recent video program "老友记 第二季 不做传统的奴隶" from YouKu.com that films a conversation between two directors, 吳念真, Wu Nien-jen (his Facebook), born in 台灣台北縣瑞芳鎮, Ruifang District, New Taipei City, Taiwan in 1952, and 林兆华 (his Sina Weibo), born in 天津, Tianjin in 1936, giving us some clues about how 儿 is used by native people from different areas.
Here are some excerpts containing 儿. If English translation is needed, I'll update it.
From 00:50 to 01:02
From 02:42 to 03:03
From 05:14 to 05:21
From 09:34 to 09:35