Can I connect ten 100W invertors (each with separate battery) parallel to power one 1kW consumer? Will it work? Is it safe? Should all invertors be equivalent or can be different?
The main problem in inverter paralleling is that the inverters must be parallel-capable. You must set one inverter as master and the others as slaves. The master has its own clock, the slaves sync to the master's clock. It also may be possible that each inverter is master/slave, so that it tries to detect signal on its output and if it doesn't detect any signal, assumes the master role. However, in that case if you disconnect the same battery from all inverters and reconnect at the same time, they may be confused due to not being able to decide which should be the master. Inverter generators are usually turned on one at a time, so there master/slave works well.
Almost always, if the inverters don't specify anything about paralleling, it means they aren't parallel capable. Making them parallel capable would mean opening the inverter and modifying the circuit. It isn't hard to someone who is experienced in power electronics, but I wouldn't recommend it for a novice.
The MOSFETs (that are the most common type of switch in inverters) are parallel-capable, so it's just that the drive circuits must be in sync.
Here's some instructions: https://yarchive.net/car/rv/inverter_paralleling.html