Why some trainers don't want to train me?
This is all because patches. If you remember, in first versions of Skyrim you could take Faendal as your companion, train Archery with him and take your money back from his inventory. Later on this was removed. Same stuff with it. Also, some NPC's are eligible to train you only if you play certain species and gender. Read the NPC's description on wiki, you'll find the full info about any NPC in Skyrim.