How to remember the non-register user choice?

Question 1

First you have to realise that all session techniques are cookie based. That is all good techniques. That means that they all have the downside that they will not work where cookies do not work user choices will be forgotten. In those (hopefully rare) cases you could store these choices either in the URL or as a CGI parameter. In any cases you can not make it really secure.

That being said, you have tradeoffs to consider.

Cookies

If you use purely cookie based storage then you could be limited in the number of user choices that can be stored in cookies under a single domain name. RFC 6265 states some SHOULDs regarding those and implementation matching these will give you at most 200KB which should be quite enough. Older RFC 2965 says implementations should give you 80KB. Also remember that the browser will send you the cookie for every request to your website. This could mean slow browsing for your users.

Assuming a 24 bits image ID (16 million possible images), base64 encoded to 4 bytes you can pack close to 20,000 choices into cookies. For 32 bits image ID, encoded to 6 bytes you still get more than 10,000 choices into your cookies.

When cookies prove too cumbersome, say after 1,000 votes you could switch the browser into the session technique… Or consider that he will never get to this without having registered ;-)

Sessions

If you decide to store the user choices in the session then you will have to dedicate some storage area on the server. The downsides are that:

you have no safe way to know when a session is not used anymore. Therefore you need some mecanism to reclaim unused sessions, typically expiring sessions after a fixed amount of inactivity,
it is more difficult to scale if and when you want to distribute the load amongs multiple HTTP servers.

Question 2

You create a unique "token" that you save as cookie (hash of IP + timestamp for example). This value is also beeing saved to the database in conjunction with the vote.