Most of the time she spoke Dothraki to the Dothraki. Especially Drogo who didn't know common. Irri of course knows common since her task was to teach Dothraki to Dany. Rakharo was obviously learning common which is evident from his talk with Jorah in episode 3.
The only place in season 1 where she speaks common in a strange place is when she adresses the remaining Dothraki at the funeral pyre. That would have made much more sense to do in Dothraki but my guess is that they kept it in english for the sake of the viewers. A similar thing happens when Jorah at the end is the one who bows down and says blood of my blood. That would have been much more appropriate for the Dothraki to say in Dothraki. Jorah saying it makes no sense.
In a blog post tied to the reairing of an episode of CNNs The Next List featuring David Peterson he talks about creating the Valyrian languages. http://whatsnext.blogs.cnn.com/2013/03/29/david-peterson-and-the-languages-of-game-of-thrones/
Specifically he has worked on High Valyrian and Slavers Bay Valyrian for this season. When translating he first translates into High Valyrian and then takes the text through a series of changes to turn it into the Slavers Bay creole of the language.
The Dothraki are a mix of a lot of steppe and prairie people which include asian people like the Mongols but there are also for example native american influences so it's not correct to say that the Dothraki are based on any specific people. Same goes for the few words that Martin has originally created. Khal is obviously based on Khan but most words have no obvious counterpart in any natural language.
Here is what was compiled by Lajaki from the first 4 books regarding the old tongue: http://wiki.dothraki.org/dothraki/Other_languages#The_Old_Tongue
Not much info there really except Jon's description and a couple words. I'm guessing there could be many ways to explain why there seems to be a style difference between the words the giants use and the longer words used by humans. It could be treated just as a coincidence since the sample size is very small. Or one could create some other historical reason for it since it's a very old language that few people speak.