Both "a University" and "an honest man" are correct. Vowels and consonants are sounds. 'University' starts with a consonant and 'honest' starts with a vowel. Try listening to the sounds instead of looking at the letters.

Questions1 and 3 can be taken together. It's not a question of how a word is wrtiten but how it's pronounced that governs whether we use 'a' or 'an'. So in the case of 'university', although the written word starts with a vowel it is actually pronounced as though it started with the consonant 'y': 'yuniversity'. Hence the use of 'a'. On the other hand, in the case of 'honest' the 'h' is silent, so the word is pronounced as though it were written 'onest'. Therefore we use 'an'. Regarding question 2, I will point you to this link: http://www.myenglishpages.com/site_php_files/grammar-lesson-articles.php Let me know if there is still something you don't understand, or is causing you problems.