How does Israel treat marriages between Christians? Between Muslims? Between atheists?
There are also a Muslim religious establishment, and a Christian one. As for atheists, it depends on whether you’re a Jewish atheist, a Muslim atheist or a Christian atheist.
Like with many things, it’s the British Empire’s fault. When they ruled here, they decided to retain the old Ottoman Millet system in which each religious community was given absolute control in matters of family law. When Israel was established the law was grandfathered in, mainly to keep Jews and Muslims out of each other’s religious affairs, and since then the various religious establishments have grown so entranced and corrupt that it’s been impossible to remove them via direct legislation - hence, all the various workarounds.
We’re in Maryland, who doesn’t really care that much. Essentially if both parties agree someone is qualified to marry them, the state doesn’t push back.
So we just had one of my friends get “ordained” online and fill out the certificate. Then we had our actual non-religious ceremony with family and friends in Virginia where my wife’s cousin “officiated.”
I know that Israel doesn’t allow interfatih marriages - but what I’m not sure about is what exactly constitutes an interfaith marriage especially if the purpose was to give religious communities control of family law. For example, is a Catholic able to marry a Lutheran?
The quickie ordination (now online but previously fax and originally snail mail) started with the Universal Life Church in Modesto California. He was nominally Christian but sincerely felt that anyone who got the calling should be able to minister and not have to go through any additional certification. His only criteria were “do only that which is right” and “everyone is free to practice their religion however they choose, so long as they don’t violate the rights or freedoms of others or the law”.
Pretty simple, right? Of course a bunch of assholes from certain religious establishments tried to sue because they’re fascists but it worked its way through the courts and the California Supreme Court correctly decided that the State can’t be in the business of deciding what is and isn’t a real religion. It’s legal now in all States. Google tells me that there are a few counties in Virginia who are pushing back lately.
What if you are an atheist who can’t be attached to any established religion? Are there no people in Israel who are atheists from far enough back that they aren’t anything else?
There is a classification of “No Religion” or “Other”, and they technically can’t marry in Israel. Marriage abroad is recognized. In addition, as I said, Israel is the land of the workaround. Israeli courts recognize the concept of common-law marriage, granting common-law spouses the basically the same rights a married couples in matters of property, inheritance, social security, child custody and so on. Many people also sign “cohabitation contracts”, which are civil agreements that creates a relationship between two people that is essentially equivalent to marriage. As with many things here, it’s complicated, messy, basically functional, and could be better.
It’s honestly the most Jewish thing ever. We have all sorts of equally arcane workarounds for Shabbat rules as well.
The first time I was married by a female judge of my father’s acquaintance. We were in midtown Manhattan on the rooftop of a hotel, and she was evidently between appointments. She didn’t put down her purse OR the plastic bodega grocery bag she had slung over her wrist! 
I once “officiated” at a fake “wedding.” The whole thing was put on by the college-age couple to fool their parents, and I was a friend of a friend that none of the others had ever met. I had no qualifications except that I was a serious-looking young man in a dark suit (and I was “from out of town”). I did it as a lark (and I think I got a few bucks) but now I kind of regret it, for reasons that are unclear to me. I guess I don’t like the idea of having committed even a probably harmless fraud.