Confusions with third-tone reading
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.