Is it reasonable to do the Trans-Siberian/Mongolian without pre-booking all the train tickets?

I did (Kyoto)-Beijing-Moscow-(Stockholm) last december.

In Russia we travelled mostly on domestic trains of various kinds (the Trans-Mongolian have some russian cars attached in Russia which allow domestic travel), using mostly "Kupe" 4-bed compartments. We made most planning and booking as we travelled.

We used the RZD web booking at http://pass.rzd.ru/main-pass/public/en and selected "ticket for own printing" (using internet and printing at the hostels). In most cases this paper is valid as a ticket when entering the train. In some cases you have to exchange it for a paper ticket, it's clearly written in english. You can then use the booking code in either the ticket vending machines, or just show it at the counter (worked best for us). Neither step needed any real Russian language skills.

Availibility : We never had any problems with availability. If you go with the domestic trains, and even more going east-west and outside summer there ought not to be any problems. RZD prices are fixed for a given day, but prices varies with the calender day (irregardless of booking pressure) and can be +20% in summer and before christmas, -5% in november and up to -20% some days in march.

Price : This was actually AFAIK the cheapest way of going. Total price for Japan-Sweden was about 700 Euro including Chinese tickets, two ferry connections, and the fact that we travelled "Kupe" in both Russia and China.

Border crossing : We travelled the trans-Machurian to Manshouli and the from russian Zabaikalsk around 10 km away. This allowed us to use domestic tickets only. Crossing the border can/should be done using bus or taxi (we tried on folding bikes which was an adventure of bureaucracy).

If you wish to go by train across the border you have to use an agency, for example Real Russia (which is also good for overview planning in any case)

China : We bought tickets for china from http://china-diy-travel.com/, a low-overhead travel agency for travellers who can do some planning on their own. They have all instructions for train travel in China and was quite fast, helpful and friendly - just try to keep in mind that they are a ticket agency - not tourist information and "all-inclusive" agency.

Stations in China were a bit crowded and bureucratic, and you had to show your ticket 4-5 times to enter your train. That said, every train was on time, we got through all the gates and the 4-bed compartments were clean and calm.

Trains in Russia were more like Western Europe, you just had to show your ticket once; to the carriage attendant to enter the car. The toilets and cars were generally clean even on non-firmery train. Going "Platskarny" may be a bit more basic; expect sweaty soldiers on the way to/from military service and a bit more chaos; but probably OK and perhaps even fun for a backpacker.

@SpaceDog gives a decent answer, but I would like to argue that it's both possible and reasonable.

If you're going to do the journey in installments, which it seems to me you're thinking of, your only option is not to travel on the trans-siberian (or trans-mongolian) only. There are plenty of train connections that travel parts of the long journey and it's these connections I mostly took myself when I did the journey (to Vladivostok, from Moscow, but not through Mongolia to China).

As an added bonus, these more local connections see much fewer tourists.

Purchasing the tickets never was an issue when I did the journey. I not once went through an agency, but simply bought the tickets directly at train stations. After I arrived in a new city, before I was to leave a few days later. Broken Russian, a dictionary and some sign language was enough to make this happen.

Proof of an onwards journey might indeed be required for Mongolia and China, but, at least for China, that requirement also depends on where you get your visa.

I traveled from Hong Kong via China and Mongolia to Moscow in August 2012 (supposedly peak season) with numerous stops on the way and without pre-booking any ticket more than a few days in advance. You must be prepared to use a few tricks though:

• Sometimes you won't get a direct route - be creative. For example, Beijing - Erlian (on border with Mongolia) was sold out, so I had to travel Beijing - Hohhot and then Hohhot - Erlian (long detour, but I did get to see an extra city)
• Sometimes you won't get the train class you want. For example, my Hohhot - Erlian train was in a "common seater car" (not sleeper), which was quite an experience (crowding, people fighting for seats, etc) but was very interesting.
• In Russia especially, if they tell you there's "no ticket" between city X and city Y on the date you want, it doesn't mean there's no space, it just means there's no single bed that's free on that entire chunk of the journey. So, for example, if there's "nothing available" from Krasnoyarsk to Moscow, there may be a bed from Krasnoyarsk to Omsk, and a different bed in a different car on the same train (or even in the same car!) from Omsk to Moscow. Ticket reservation agents don't know to look for this, you must ask for it yourself (or better yet find such combos online on the RZD website if you speak Russian)
• In Russia, use a service agency to book train tickets (there are usually a bunch of them in the center of any city) instead of buying at the train station. Yes, they charge crazy commissions, but it's worth it, you can buy tickets in a much more relaxed atmosphere, can explore your options, etc, and there's probably a better chance of finding an English-speaking agent
• Obviously, only attempt this if you have a few days of buffer in your schedule, but in general don't be afraid of getting "stuck". Just take the next available local train to the next major Transsiberian city, even if it's not the one you want, and you'll eventually get there - these local trains run very often. And if you really get stuck, there's always planes.
• It may be more difficult to get last-minute tickets to "international" trains (ones that cross Russia/Mongolia or Mongolia/China borders), but keep in mind that you can always take the train right to the border, cross it by land and continue by train. I did this when crossing from China to Mongolia (for cost savings reasons, international trains are overpriced) - there are jeeps that take you across the border for a reasonable fee to the train station on the other side.
• From what I saw, Russia was the best when it comes to last-minute tickets, China was probably the worst, with Mongolia somewhere in the middle. In Mongolia, don't trust what you're told at hostels/guesthouses, get the tickets at the train station yourself if you can (in addition to not having complete schedule information, guesthouses sometimes mark up the price by 50-100%)

Unfortunately due to political reasons train transportation in Russia is getting worse as far as I know. My data is 2 years old, apply it to the current situation at your own risk : )