Buying ticket on Russian train

No, you can not do this, at least on most trains and as long as you want to do everything officially. You will need to go to a ticket office and buy an additional ticket for the needed segment. There are many reasons for this, but one is very simple: the conductor simply has no means to find out whether there are any tickets left on the second segment, or the carriage (or train) is already completely booked.

There are few trains in Russia where the conductors are equipped with mobile ticket selling terminals (which do check seats availability in particular), but these are the "best" trains (Sapsan and may be a few others), I really don't think that on the train to Zabaikalsk the conductors will have this equipment.

You might try to do this unofficially, that is actually bribing the conductor. I even guess that this may work in some cases. However, this is definitely illegal. I think then in general case the worst consequence for you can be that you will be just thrown off the train at some intermediate station and required to pay a fine. For the conductor, this may be something more severe. But in your particular case you are crossing a state border, and not having a proper ticket can bring you much more problems with the border control.

Talking about buying a ticket while already on train in general, well, technically you can go to a ticket office while your train stops at some big station. However, the lines at the ticket offices are usually rather long, so I would not rely on this. You can also buy a ticket online, but then you will still need to go to a ticket office to get your ticket printed on an official form. You can still find a long line to a ticket office, and finally find yourself both without money (as you already paid for the ticket) and without a ticket (as you did not have time to print it). Note that "electronic registration" option, which allows you to board a train without an officially printed ticket, is available only if you buy your ticket at least 1 hour before the train departs from its starting station, so that's not an option for you.

On big stations there are terminals where you can both print a previously-online-bought ticket or buy a ticket anew paying by a credit card. However, these terminals are not always working and the lines to them might be long too. So the terminal option is a bit more reliable than manned ticket office, but still not very reliable. I would not use it unless you train stops for 40 minutes at least.

I would think that it might be possible.

It is possible to upgrade your class: if you have platzkart, you may upgrade to kupe while on board. Therefore, they do know on board which seats are empty and which are available.

So, you might be able to do that, seek out the "director of the train" (yeah that's a hilarious title). It might also be available unofficially if train is less than full, ask conductor in your car carefully.