Protecting JavaCard application and data memory

Is it possible to protect JavaCard application against program copying from one card to another? Also is it possible somehow access data on JavaCard and damage them?

Sure, generally you cannot retrieve Java Card applets at all. You can load them using Global Platform (GP) LOAD commands and install them using INSTALL. There are however no retrieve commands as far as I know, nor are they necessary. If they were present then you'd still need the GP key set to install them. If you have a card with a default key set then you can replace the keys before or after loading the applet in your secure (pre-)personalization environment.

Usually the chips are protected e.g. using a hardware mesh and / or similar passive & active protections to avoid extraction using hardware. In the end though you should use the applets in such a way that an attacker would have a hard time getting information out even if the code is known (using derived, card specific keys for instance, calculated or loaded during (pre-)personalization).

Good cards are tested against e.g. Common Criteria. CC does publish reports and those should be referenced by the relevant product pages. Of course, that doesn't mean that they will never be hacked - the adversary may have direct access to the hardware after all, but it does provide a minimum level of security.