I'm trying to understand the timing priority chart. I see that on the Runner's turn, after each action, non-ice cards can be rezzed. So after a Runner's click, the Corp can rez any non-ice?

Yes. Also make sure you use the new timing chart from the FAQ as it has some updates from the one in the rulebook.

When would it be the right time to rez a card during the Runner's turn? I read too that if this happens, the Runner acts first if he has paid abilities.

Are you reading this from the FAQ or the rulebook or online somewhere ? I don't know what you are asking by "when would be the right time" but the FAQ has the timing chart on page 5 to show you when things can happen.

 Let me reword that, I meant, when is it good to rez a card on the Runner's turn? Like a certain strategy for this, can't seem to find the point in doing so.

But, I have another similar question…

The Runner has installed a completely made-up card:

5 credits: The Corp discards one card from his HQ

It is the Corps turn. On his second click, he draws a card. The Runner has 12 credits and uses the paid ability twice and the Corp discards his hand down to 1 (3 cards in his hand).

Now, would this seem right as if the Runner would use paid abilities on the Corp's turn, and like if the Corp would rez cards on the Runner's turn? 

…Rulegeeking here.

Don't see the problem. Triggered or paid abilities can be triggered whenever the timing chart allows for that (always with the player whose turn it is having the first go at it).

As for when to rez a card on the runner's turn: You can't answer it that easily. If you want to scare the runner of from proceeding on the run, rez it as soon as possible. If you want to spring a trap do so just before he/she accesses the server.

 I like to rez assets like PAD campaign or Adonis during the runners discard phase. Of course this can also be done during the corps draw phase. The reason is to hold onto credits during the runners turn to rez ICE or pay for traps. Finding the best time to rez cards is an important part of corp strategy.

Sneaky - I hadn't thought of that.  My thinking process when I first saw that card was "Oh cool - creds at the start of my turn - I'd better rez it now in order to take advantage next turn."