What should the input mpedace of an antenna be, 50 ohm or other values?
If they do so, the antenna will NOT resonate at the frequency the antenna is matched. The goal of matching is to transmit/receive as much power as possible. In this case, it is a receiving antenna. So, will the antenna performance be lower than it would be if it resonates? 


Well, you kind of self-answered that: you didn't do a power matching, so not all power can be used by the rectifier. I'd call that worse performance, yes.

Not quite sure that you've got a reasonable idea of what the antenna does: Your "it doesn't resonate" statement imho is incorrect, and might hint you're confusing a microwave patch antenna with a e.g. HF loop antenna where you "match" with a capacitor to achieve resonance between the loop and the capacitor; this is nothing like that, there's no voltage or current wave coming back from the matching circuit to the antenna to "resonate". That's the absolute reason to have that matching circuit!


  Usually the antenna is tuned to be 50 ohm at the resonant frequency. Is it better to do so, compared with the case in question 1?


I don't understand the question. You need to match the impedances of all components in the system. 50 Ω is a commonly used impedance, but any real-valued impedance could work well – you just need to make sure all components (and especially the transmission lines between them) have that impedance. (Complex impedance matching would work, too, but then you'd need to consider material losses more intensely – something easy to do for someone in a research lab with experience and access to special PCB substrates, but not as fun for you with what I presume is a FR-4 patch antenna idea).

In your specific system, I'd argue that 50Ω is not a good choice. Build a matching circuit that converts your rectifier's complex impedance to a real one, and then modify the design of your patch antenna (which inherently has design parameters that allow you to set the antenna impedance – read up on why your patch antenna has its feed in a specific position with specifically deep "ridges" where the feed enters the patch).

Not much magic here – you need power impedance matching. So do impedance matching to the same impedance. Pick that impedance as convenient, if your really only have the antenna and the rectifier in your circuit. If you've got more than two, pick a "sensible" impedance to match all components to – 50Ω is a typical choice for that, but it really depends on what you're dealing with.


  In practice, do I need a balun when measure S11 / input impedance of the antenna or of the rectifier by a network analyzer? My PCB antenna is a PIFA one, so it is an asymmetric one. 


That makes no sense – you want to rectify, so balanced signals are undesirable in every thinkable way.