is there maximum of live wires allowed in one conduit
There are two conduit factors:

Thermal derate

Yes, overheating is an issue, but the limitations don't really start to pinch until you have more than four circuits.  

Generally in a split-phase system, the rule of thumb is 2 wires per circuit (a circuit being something powered from one breaker where all handle(s) throw together) ... unless for some reason current is making 2 round trips down the pipe.  

More precisely, you count the number of "wires" with some exceptions.  Grounds are not counted. In residential split-phase, neutrals in MWBCs or 120/240V circuits are not counted, because current flow on them means less current is flowing on another wire.  

Broadly, wires which split current from the same breaker are counted as one.   In a fan+lamp circuit where one hot feeds two switched-hots, the two switched-hots count as one wire.  Something to think about if you don't actually have 6 breakers feeding this. 

If your adjusted wire count in a single pipe &lt;= 9, no de-rate is necessary for 15-30A branch circuits provided you are using 90C rated wire (THHN or THWN-2 or XHHW-2 or several others).  

If you have 10-20 active wires, you must de-rate the wires as follows:


wires on a 15A breaker must be #12
wires on a 20A breaker must be #10
wires on a 30A breaker must be #8 Cu or #6 aluminum.  


In turn, the larger wires require larger pipe.  

Conduit fill

For this, you can use any common web calculator.  You appear to be fine for 12 wires in 3/4” EMT up to #10 wire.