Why do almost all international trains to Russia show as sold out?

There are several things going on here:

Popular summer travel

Travel in July and August is popular; there are likely many Russians on vacation in Europe.

Trackworks

Specifically for the Paris-Frankfurt-Berlin-Moscow train, there appear to be track works in France in summer. Tickets between Paris and Berlin are not available beyond 15 August. When looking at the train Berlin-Moscow for 16 August, RZD reports:

Временно закрыта продажа билетов на участке Париж – Берлин для
поездов, отправляющихся из Парижа с 15 августа. На участке Берлин –
Москва продажа осуществляется в обычном режиме.

And the arrivals and departures at Berlin Hbf show that the Moscow-Paris train is shortened between 5 July and 9 August and
between 14 August and 6 September, running only as far west as
Offenburg (westbound arrival Offenburg 03:51 or 04:56, eastbound
departure 23:39 or 0:18). When searching through bahn.de / HAFAS the
train shows up normally from Paris until 8 August, but not from 15
August, when HAFAS also shows it as running only from Offenburg:

So it looks like there is something going on (I guess
trackworks on France). The train may not be running to or from Paris
on those dates at all (I’ve read before that the English language
interface at RZD shows trains as “sold out” when they are unavailable
for booking for any other reason, such as when searching too far into
the future, so that is consistent), or even when it will, by the time
RZD opens the Paris-Berlin section for booking the train may be sold
out between Berlin and Moscow! So it looks like travellers in this
period may need to travel on different trains to Berlin then get on
the train to Moscow there.

Tickets available, but not (all) through RZD

For the Warszawa-Moscow train, there is yet another message from RZD:

Уважаемые пассажиры! Для продажи на сайте ОАО «РЖД» в поезд №10 Варшава-Москва «Полонез» предложено 12 мест. Продажа остальных мест на данный поезд осуществляется в международной билетной кассе. При отсутствии свободных мест на сайте, просьба обратиться в международную билетную кассу.

which appears to mean that RZD only sells a limited number of tickets for this train. Indeed, as shown by the screenshots in the question, even when RZD reports a train as sold out, Polrail may still be able to sell tickets.

Tickets made available later

I have checked the departures for 2 August, 9 August, and 16 August again. When I checked those on 22 June (5 days ago), they all showed as sold out. Checking them again today (27 June), there are (in second class) 7 seats available on 2 August, as well as 43 on 9 August, 49 on 16 August, and 39 on 23 August. So it seems that some tickets are made available only later — perhaps extra carriages were added to trains that had sold out.

Traveling to or from Russia by train is neither time nor money efficient, what with the different gauge system and the lack of proper high speed rail infrastracture anywhere between Moscow and Brest.

You can easily see it yourself by comparing travel times for the Paris-Berlin-Moscow train. The cheapest ticket you could get from Berlin to Moscow would cost approximately €160, with the trip taking a whopping 24 hours. Compare this to the cost of the cheapest non-stop airplane ticket of €90, with the flight taking less than 3 hours.

Traveling abroad by train is and has always been a luxury tourist attraction in Russia. As such, the wait line can get pretty long throughout the vacation season when everyone wants to get the fun train ride.

One could make an argument that with increased demand there should be greater supply. One reason that is not the case might be the fact that there is only one state-owned rail company operating in that region, yet there are multiple fully competitive airlines that operate in Russia.

The only fast and affordable train connecting Russia with a non-Soviet bloc country I can think of would be the Helsinki-St. Petersburg train. Note that the train is operated by the Finnish VR group (as opposed to RZD). A seat in the second class car would cost mere €39, getting you to your destination in just 3 hours. Moreover, you could book it for almost any day, even tomorrow.

To summarize, the likely (though admittedly unsatisfying) answer to your question “why do almost all international trains show as sold out” would be “because the Russian government wants it this way, or just does not care enough”.