Not exactly
While he certainly serves as a moral compass for <PERSON>, <PERSON> isn't the moral conscience. <PERSON>'s morality is complex - just like yours and mine - and comes from lessons his parents taught him, how his parents died, his upbringing under <PERSON>, and a myriad of other factors. This question helps unpack <PERSON>'s morality better than I can:
http://scifi.stackexchange.com/questions/87283/why-didnt-batman-just-kill-joker-when-he-had-the-chance
<PERSON>'s role is mostly that of a caregiver, technical assistant, field doctor, and manager of his estate but along with the rest of the <PERSON> Family, he helps keep <PERSON> from letting his crusade against evil take him too far down a dark path.
DCComics.com states:
But <PERSON> had a far different way of honoring his parents in mind, and when he adopted the cowl, <PERSON> came along for the ride, tirelessly guarding his employer’s secret while providing some much-needed assistance. Whether it’s skilled technical support, medical aid or simply managing the complicated schedule that comes with being a business leader, nighttime vigilante and wealthy playboy—it’s all under the purview of Mr. <PERSON>. Driven by love and an unwavering sense of loyalty, <PERSON> nevertheless worries deeply about <PERSON>. He understands the importance of <PERSON> to a city such as Gotham, but after a lifetime of raising <PERSON>, he still sees a young, scared boy where most see a hardened hero. 417
And every night the bat-signal flashes across the city skyline is another night that boy will spend searching for a way to undo something that could never be undone. A search that <PERSON> realizes could very well lead to his end.
To answer your sub-questions:
1. You'll find examples of <PERSON> giving <PERSON> advice all the time - most of which isn't moral in nature but rather parental or caring. Panels such as this one pop up pretty regularly:
1. <PERSON>, <PERSON>, the rest of the Batman Family, and <PERSON> in some instances all give <PERSON> moral advice but it's usually more about letting people in, trusting his team, and not overworking himself rather than simple matters of right and wrong.
2. No. Most iterations of <PERSON> use the <PERSON> persona as a show or front to disguise the "real" side of <PERSON> that has devoted his life to crime fighting. To describe them as two separate entities implies some sort of split personality disorder that simply doesn't exist. 494