Confusions with third-tone reading I came across a text. It is about shopping at a 商场. In Taiwan, I believe it's shang1 chang2, but in Mainland China, it's shang1 chang3, and I want to go with that one, however, it is causing me confusion.

(1) So there was this phrase "商场里", as in "自选商场里的东西真多"; the pinyin annotation was shang1 chang3 li3; now, I know that there are full third-tone （which I'll refer to as full-3) and half-third tone (which I'll refer to as half-3), so when I read, or when I teach someone to read, is it (A) shang1 chang(half-3）li(half-3) or (B) shang1 chang(2) li(half-3)? Or is it something else?

(2) And then there's this sentence: 收款的阿姨用电脑很快算出了要付的钱。“用电脑很快” is the part I am having trouble with. The pinyin annotation was yong4 dian4 nao3 hen3 kuai4, so is it (A) yong4 dian4 nao(half-3) hen(half-3) kuai4 or (B) yong4 dian4 nao2 hen(half-3）kuai4? I suspect (B) is wrong in this case, but I have a hard time justifying to myself why (A) is right. I believe (A) is right because you can separate the sentence like this, 收款的阿姨用电脑 // 很快算出了要付的钱, so because 脑 is part of 电脑 and 很 is part of 很快, both can be read in half-3 tone. Does this argument hold?

(3) Speaking of which, it has been a very long time since I have been in weekend Chinese school, and weekend Chinese school is different Western schools' Chinese course (which I have never attended) in terms of teaching styles and materials, so I am wondering, now in Chinese education, when students see a 3-tone character by itself or at the end of a binome consisting of 2 3-tone characters, are they taught to read AND speak in half-3, or are they taught to read in full-3 and speak in half-3?
This is a very common tone sandhi pattern in Mandarin. When two "rising" tone (上聲), i.e. tone 3 in Standard Mandarin, characters follow each other in a word, the first character is changed into "light rising" (陽平), i.e. tone two.  

So, 商場裏 is pronounced shang1 chang2 li3. However, (2) is a bit contendable, since this sentence should be broken down into 用電腦/很快. Some people would say yong4 dian4 nao3 hen3 kuai4, and some would go with yong4 dian4 nao2 hen3 kuai4. I find myself using the latter most of the time, but either way should be comprehensible.

No, native Mandarin speakers don't usually recognize the difference between half-3 and full-3. I find myself pronouncing words with half-3 most of the time, but sometimes full-3, without myself recognizing it at the time.