As Jews with the Holocaust just 80 or so years behind us, we should be extremely sensitive to civilians being targeted with  nowhere to go. We should take in Jews and non-Jews alike - anyone who has no place and needs to get out of the war zone should know that there are countries who will take them in. Whether there are richer countries or closer countries who should take more people in is not our problem - we should accept anyone needing a refuge. We know what it is like to be stuck in a place of war and danger with no country opening its doors for us. 

All issues can be resolved. Jewish, and probably some non-Jewish refugees can come under the law of Return if they want to move to Israel permanently or even just as a temporary safe haven and leave when things quiet down or when they find somewhere else to go. Non-Jews and those who don't qualify under the Law of Return can be made aware that we will give them refuge until things calm down, for x amount of time.

The State will not lose its Jewish character by taking in some non-Jewish refugees for a short period of time, even if it turns out to be a few years (which it probably won't) - it will lose its Jewish character if it does not. 

No, maybe we don't have to. We are not a bordering country nor are we a particularly wealthy country. But if nobody else is opening their doors for the refugees we have to.