Cannot ping from a bridged inteface

That is the intended behaviour. If you connect eth2 to a bridge then eth2 becomes unusable (even with an IP address). You use br0 on the software side, eth0 is used physically.

In general when talking about bridges it makes sense to give the output of brctl show

There is no ipaddress on interface eth2.

This is also the problem.

This is as it should be. Ping is a layer 3 thing. A bridged interface is a purely layer 2 thing. You can't ping from an Ethernet cable either for the same reason. Just like a pure switch has no idea what ping is, a pure switching interface has no idea what ping is either.