Transport from the center of Moscow to DME is quite easy. From the Paveltskaya train/metro station there is an express train. Take a look at the Airport website, domodedovo.ru. There are train schedules online and also it should tell you whether your plane tickets gets you a free ride or not.
It won't be easy to guarantee getting from Suzdal to Moscow on a Friday.
There are two possible ways to travel. One is the bus from Suzdal to Moscow. It takes about 4 hours. The bus starts in Ivanova and I am not sure that you can prebook tickets which means there may or may not be seats available. To make your plane, your bus would need to arrive in Moscow you would need to board the express train about 1500-1600.
The other way to get from Suzdal to Moscow is to take the bus to Vladimir and the train from there to Moscow. It takes a little longer but you have a better chance of being sure of getting a seat.
Ruth