Can I fully load multiple breakers (rooms) in my apartment?
I don't know about your local code requirements, but in almost all countries at least the building has some breaker to connect to the power grid. 

For example for my house this means \$3 \cdot 63A \cdot 230V = 43.47kW \$. Check it for your house. That is a whole lot of power that could easily set lots of things on fire.

However there are some things to note that do not make all this available for me:


There are other persons/electric devices in my house
the power company allows at most 40kW to be drawn continuously (due to limitations of the installed measuring equipment)
Drawing so much power will lead to a voltage drop that depends on many factors.
Most breakers do not like to be run on their exact amps value and trip after a while (sometimes hours). Use some safety margin.


Depending on the wiring you use (inside the walls and as short, non chained extension chords), they will introduce some more or less noticeable voltage drop, further decreasing the usable power for your devices. 

For safety reasons you should check (or if you do not have the tools and knowledge to do so, let it be done by someone else) all equipment involved. There could also be other limiting factors involed, e.g. an RCD/GFCI.

125MLO indeed means that the panel is rated for max 125A.

For extra safety install multiple kill switch methods and a smoke alarm...