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"?
Answer to each question
"When Beijingers use words like…"
Assume that by "Beijingers", you mean those who born in or near Beijing, so do their parents or close relatives, growing up in a Beijing native people community, and not necessarily living in Beijing all the time of a year.
"When Beijingers use words like 一点儿, the ending syllable will be very pronounced as far as I understand, I'm aware that this is a typical feature of the Beijing dialect."
It is a feature for local native people around Beijing area, from a native speaker's viewpoint, not because the word 儿, but because their pronunciation, that is, their articulated way of the r consonant. For theory behind it, refer to others' related posts.
"But how is it pronounced in the rest of China?"
Take 一点儿 for example, I can't give an absolute answer, because (1) I can't guarantee that the rule, if any, applies for each and every native speaker, (2) there seems to be hardly a theory that can catch up with real life conversation. But I can tell that both 一点儿 and 一点 can be used in some areas, such as Chongqing, Wuhan, and Shanghai. Few people (that ain't Beijing local native speakers) tend to stress on the tailing 儿 sound, though I can't say never. Imitating Northern Mandarin can be fun for some people too.
"Is there no 'er' sound there at all, or is it only there for special cases 'e.g. 一点儿'?"
Continued with the above 3., 一点儿 is also a little bit special case too, that is, this phrase is so common that some people, including me, may feel it kind of odd without using 儿. 一点 may be used for normal more often than 一点儿, however, 一点 is not so formal that would be used in formal cases such as in a speech of a conference, instead, we have 些许, 稍微, 少量, 片刻, to name just a few, to express ourselves in some formal situations.
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
(原字幕, Subtitle in the video) 排戏都太严肃了 正经八百地 你还得导演阐述 这个戏剧的主题是什么 这个主题要飞跃到一个什么高度 什么他妈高度 再高把你摔死了
(我的簡體中文聽寫稿, My transcript in Chinese Traditional) 林兆华: 排戏都太严肃了 正儿八百地 啊 你还得导演阐述 这个戏剧的主题是什么 这个主题要飞跃到一个什么高度 什么他妈高度 再高把你摔死了 (Here, 正儿八百 is more casual than 正经八百, meaning "seriously")
From 02:42 to 03:03
(原字幕, Subtitle in the video) 有时候觉得 每天早上醒来的时候 就会 一点儿把握也没有 都会觉得说 好不容易睡着了 干嘛又把我弄醒了 然后再去面对那些烦恼 一次又一次 如果你是我 你会希望再醒过来吗
(我的繁體中文聽寫稿, My transcript in Chinese Traditional) 簡南峻(吳念真 飾演): 有時候覺得每天早上醒來的時候 就會 一點把握也沒有 都會覺得說 哇好不容易睡著了 幹嘛又把我弄醒了呢 然後再去面對那些煩惱 一次又一次 如果你是我 你會希望再醒過來嗎 (電影 "一一" 片段, from "YI YI" the movie. Here, the speaker uses 一點 (i.e. 一点) instead of 一點兒 (i.e. 一点儿), though the subtitle has 儿)
From 05:14 to 05:21
(原字幕, Subtitle in the video) 老娘爱看戏曲 就有时候看点评戏 看点京戏 就跟着去看 就是这点基因 我父亲爱拉弦 拉胡琴
(我的簡體中文聽寫稿, My transcript in Chinese Simplified) 林兆华: 老娘爱看戏曲 就有时候看点儿评戏 看点儿京戏 就跟着去看 就是这点儿基因 我父亲爱拉弦儿 拉胡琴的这这 (Here 儿 is used for four times.)
From 09:34 to 09:35
(原字幕, Subtitle in the video) 你跟他较劲
(我的簡體中文聽寫稿, My transcript in Chinese Simplified) 林兆华: 你跟他较劲 (Here 劲 is used instead of 劲儿. 你跟他较劲 means "You take on him. You compete against him or start a fight with him, especially when he is bigger or better or stronger than you.")
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.
抠门儿 kōumér – stingy, miserly (may be used even outside Beijing)
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.)