Christine, I hope following answers would help you and others.
Which kinds of accommodation are available in Mongolia?
Flats, apartments, detached or terrace houses. Town centre, or pristige area of Zaisan, Tuul river area.
Is it hard to find a flat or a house for rent in Mongolia?
If you are a foreigner, finding a accommodation by yourself won't be easy. Besides the language barrier, you might have misunderstanding what to expect, what to pay, and the liabilities.
How to find a home in Mongolia: classifieds, real estate portals, real estate agencies?
I would advice to consult with agencies. They won't charge you for finding an accommodation, unless you purchase a follow up services.
How much is the rental for a house, an apartment?
depending on the area, furnishing, and environment, houses range- 2000-8000$ p/m. Apartments flats, 900-3500$ p/m. Usually rent quarterly, but lately some landlords prefer half a yearly.
What are the formalities or the required documents for renting an accommodation?
Employer reference, passport copy and rental agreement. One months deposit to be paid and all rents paid in advance.
Which advice would you give?
Better consult with real estate agents as landlords are not very experienced. Do not hesitate contacting me for further advice, [Moderated]