Part 2:
The Mandarin in mainland China and the Mandarin in Taiwan are the exact same language, with only slightly different accents; however, Taiwan uses Traditional characters while China uses Simplified characters to write the same language. Cantonese is a completely different language from Mandarin, but if you go to Hong Kong you'll see that Cantonese is written in Traditional characters, exactly like the Mandarin in Taiwan. When I was doing a Cantonese course last year I practised with native speakers from both Hong Kong and Guangzhou. Not many people in Guangzhou can actually write Cantonese, despite being native speakers, but those I practised with wrote it in Simplified characters i.e. they wrote Cantonese sentences with Cantonese grammar, but used simplified characters to do so.
Keep in mind that Chinese characters are non-phonetic. This means that cognates between Cantonese, Mandarin and Classical Chinese will use the exact same character, even if they're now pronounced completely differently or they mean something else.
Long story short: Mandarin is a language (like English and Norwegian), which can either be written in Simplified and Traditional characters (depending on which country you go to). Before, I described the difference between the two scripts as being like the difference between Latin and Cyrillic, but I think it'd be more appropriate to compare them to fonts. It's sort of like the difference between clear, printed letters and highly stylised cursive handwriting.