How practical is it to visit northern China and Mongolia on a

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How practical is it to visit northern China and Mongolia on a trip to Southeast Asia? I'm in the rough stages of planning my destinations for a trip to Asia and I'm trying to figure out how practical this would be

>Budget $5000
>3 months from early May to early August

>Fly to Beijing from the USA
>Great Wall, Forbidden City, all the normal tourist stuff
>Trans-Siberian train up to Ulaanbaatar
>Spend a day in the city
>See the Gobi Desert, see the Steppes, go hiking somewhere
>Train back to Beijing
>Work my way south towards Thailand/Laos/Cambodia/Vietnam
>Might fly directly from Beijing, might stop somewhere along the way if it's worth visiting
>Spend the bulk of my time in SEA
>Head to Malaysia, Singapore
>Fly to USA from Singapore

Is it just too far away from SEA to go to China and there? I'm interested in pretty much everything - history, nature, beaches, foreign culture, food, nightlife, meeting other backpackers - so I want to go to where I can get the most diverse and interesting experience while I'm abroad.