No they can't ping each other because they are on different subnet, a router is require if you want to. In a VLAN situation it's the same, they won't be able to ping each other without a router. Plus you can't take a class A address of 10.10.10.1 and reuse it as a class C 10.10.10.1 on a network with that scheme it won't work either. Because the network address 10.10.10.0 is taken by the class C address so the class A address will conflict each other. If you ping 10.10.10.1 on the class A side than what you're really doing is a loop. So your answer is yes you can ping 10.10.10.1 but no you can not ping the other side 10.10.10.1
Confused?
Let say we have a switch and 2 host connected to it, no vlan setup.
Host A address is 10.10.10.1 255.0.0.0
Host B address is 10.10.10.1 255.255.255.0
When Host A pings 10.10.10.1 it is pinging itself! So yes you can ping BUT
If I change Host B address to 10.10.10.2 255.255.255.0
When Host A pings 10.10.10.2 it times out so no you can't ping on different subnet without a router.