'anime' is short for 'animation' in Japan, which refers to, you guessed it, animation of any kind. They call western cartoons anime just as they call their own cartoons anime. They also call 3D movies anime (For instance, Shrek is an アニメ映画). While 'anime' has been readopted into English as a way of referring specifically to Japanese (and sometimes other Asian) animation, they're still considered to be a subset of animation and cartoons, and both terms still apply.
Just like we say 'manga' to refer to Japanese comics, but the word 'manga' just means 'comics', and is used for comics from any part of the world in Japanese. Again, while 'manga' has been taken into the English language to refer to specifically Japanese comics, manga is still a subset of comics, and 'comic' still applies, except to kids yelling "Mooom! They're not comics! They're MANGA! And these other ones are GRAPHIC NOVELS! I'm an adult!"
Anime is just as much Cartoon as Family Guy, The Simpsons, and King of the Hill And the little kids anime is just as much anime as Ghost in the Shell and Tokyo Godfathers. There's no age line
Doumo is like 'very' or 'much' or whatever. It's often used to imply the 'thanks' part by itself. It's almost set to the whole 'thank you' expression, but you do see it in other places every now and then.