Hi - you intend to travel on the Trans Mongolian (the Trans Siberian goes to Vladivostok not Beijing) - as mvbergen says check the man in seat61 website -- there are a few agencies, most well known is Real Russia, however allagencies charge a fee obviously. It is fairly easy to book tkts yourself for train but you will need help from some agency re visas / letters of invitation (eg https://www.expresstorussia.com/russian_visas.html) --- easy to get around Russia / Mongolia and China solo

Trains like this in Siberia don't sell out months or weeks in advance.

Yes of course I overthink, that’s my style! Ha. I travel a lot & I have found overthinking has its pitfalls but has avoided some situations, for me, that would have been unacceptable. In any event,that is exactly my plan..do one leg of the trans Siberian by train & perhaps fly another leg.

Yes it is, at least, in my opinion. I always prefer (with the exception of Trans-Siberian trains) 3rd class, so called "platzkart" as Russians call it. Platzkart is better for a visitor, you can socialize with other passengers, so there is no real reason for paying more to be restricted to a compartment. Sometimes, it feels so sterile to travel in this class.

By the way, I encourage you to try both classes because I would like to hear your opinion when you return home from Russia.

I did not use this outfit, but they have been around for awhile:

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Monkey Business Info - Trans-Siberian since 1988

We specialise in Trans-Siberian tours through Russia and Mongolia. Start your trip in Beijing, Moscow or Vladivostok - travel the entire Trans-Siberian or Trans-Mongolian route, or depart from Beijing on a round-trip circuit through Mongolia and Siberia. Whatever your travel style, time frame or budget, we can help plan your next adventure.

Well I guess you won't be hitching in Siberia this year at least. I just got back from Yakutia last week. The coldest it was was -37°C. This is what I wore. T On top: Merino wool inner layer, polar fleece outer layer, down parka with substantial hood. Bottom: Stretch pants (thicker than leggings) outer windproof ski trousers (thicker than nylon). Fleece neck-up, woolen hat with ear flaps. Thick insulated ski gloves (I actually had two pairs of gloves but never wore the inner layer. Thick wool hiking socks. Sorrel type boots with felt lining. I got a bit cold a couple of times when I took my gloves off to take photos but could quickly recover by putting them back on.

I thought that this question was a classic troll, but, as indicated below from Wikipedia, times have changed. I took the Trans-Siberian Railway in the 1980s, and I have driven the Alaska Highway several times, and did not believe that there was an all weather road, yes driving on snow, across Siberia.

Are there filling stations every 200 or so KM with gasoline? I assume that diesel would be like tar in January. Do the filling stations have truck stop cappuccino? And heated places to sleep?

START WIKIPEDIA PASTE JOB:
The route, in places coinciding with European route E30 across a distance of about 190 kilometres (120 mi). One of its segment (Chelyabinsk-Novosibirsk) can be passed by the R402 highway via Ishim inside the Russian territory or by the R254 highway through the neighboring country of Kazakhstan. The route consists of seven federal highways:
M10 Russia Highway: St. Petersburg-Moscow, 664 kilometres (413 mi)
M5 Ural Highway: Moscow-Chelyabinsk, 1,880 kilometres (1,170 mi)
Baikal Highway:
R254: Chelyabinsk-Novosibirsk, 1,528 kilometres (949 mi) or 1,630 kilometres (1,010 mi) with R402 bypass highway inside the Russian territory.
R255: Novosibirsk-Irkutsk, 1,860 kilometres (1,160 mi)
R258 Baikal Highway: Irkutsk-Chita, 1,113 kilometres (692 mi)
R297 Amur Highway: Chita-Khabarovsk, 2,100 kilometres (1,300 mi)
A360 Lena Highway: Major branch leading to Yakutsk and northeast Siberia, 1,235 kilometres (767 mi)
R504 Kolyma Highway: Extension of A360 to Magadan, 2,031 kilometres (1,262 mi)
A370 Ussuri Highway: Khabarovsk-Vladivostok, 760 kilometres (470 mi)
Amur Highway[edit]
Main article: R297 highway (Russia)
Until 2010[2] the most problematic stretch of the highway was between Chita and Khabarovsk. The first section of this route, linking Belogorsk to Blagoveshchensk (124 km in length), was constructed by gulag inmates as early as 1949. Extended and updated between 1998 and 2001, this road forms part of the Asian route AH31 connecting Belogorsk to Dalian in China.
The Chita-Khabarovsk road remained largely unfinished up until early 2004, when Russian President Vladimir Putin symbolically opened the Amur Highway, with great swaths of forest separating major portions from one another. Jim Oliver and Dennis O'Neil rode motorbikes across Russia, along the Trans-Siberian Highway, during the last week of May and the first three weeks of June in 2004:back then, as described in Jim Oliver's book, Lucille and The XXX Road, the section between Chita and Khabarovsk was an extremely challenging undertaking among marsh, gravel, rock, mud (vulnerable to the rasputitsa seasons), sand, washboard, potholes, stream fording and detours of the elusive highway with a noticeable absence of pavement which leads into cases of probable surface tension which can cause the highway to collapse. In the following years the road, in some places was a modern paved highway with painted reflective lane-lines, while in others a single lane meandering, pockmarked, loose-gravel trail following the route of the early 20th century Amur Cart Road. Completion of a 7-metre-wide highway between Chita and Khabarovsk was slated for 2010: now the road is in very good condition, completely upgraded and enlarged and with a smooth surface.[3] The Amur Highway was fully reconstructed and paved in September 2010.