When two people get married (1- hetero couple and 2- homo couple), lets say in the US, how is it recognized internationally? Let say the couple travels to Europe, are they recognized as being married? Or do they need to carry around "proof" of marriage? What kinds of documents need to be acquired to be "legally" married everywhere? Is there even such a thing? Is it very different for gay as opposed to straight couples?
There are still a handful of situations where being married makes things possible that would otherwise be impossible. (Hospital consent, hotel room sharing, being on the same immigration form, etc.) In the vast majority of these cases, an opposite sex couple can get through by simply stating they are married. Occasionally the "proof" of having ID with the same last name is needed. Where things get tricky is when the couple is the same sex and/or has different last names.
Don't underestimate the power of simply stating things that shouldn't be believed. I have been claimed as a sister more than once because of rules (about how many families can camp on a single campsite, and who can visit whom in the hospital) and nobody asked for proof - even when the two sisters had the same first name, or another time when the two sisters were of different races. If you are married and you say so with confidence it's unlikely you'll be asked for a marriage certificate.
Unfortunately that can work in the other direction too. If you're blocked from doing something, but you're married, and you're dealing with someone who doesn't want to let you do it, you may be out of luck. Even if you produce a marriage certificate, some nurse or hotel clerk or immigration officer can simply announce that "your marriage isn't recognized here" and there won't be much you can do about it. (You could maybe sue later, but I am talking about getting through whatever the block is on the spot.)
Technically, most countries recognize each other's opposite sex marriages but possibly not same sex ones. For example, a Canadian government page about getting married outside of Canada says:
Marriages that are legally performed in a foreign country are usually valid in Canada, and you do not need to register them in Canada.
It then goes on to warn:
Although same-sex marriages are legal in Canada, they are not recognized in many countries. Same-sex civil unions are more widely recognized.
There is no international standard for this as there is for passports, probably because marriage customs and procedures are just too diverse.
As Kate said, people will usually take your word for it for most everyday purposes such as travellign (getting a visa for Saudi Arabia may be the exception). Actual proof of marriage is usually only required for things like immigration, taxes, inheritance and child custody. The demands there vary as well. Sometimes an official document like a marriage certificate or marriage license may be enough, in other cases you even have to prove that the marriage was consummated.
That would entirely depend on the countries in question. For many there's no reason to care about it at all, unless you're planning on getting married in a country where you're not residents.
For example if you're Dutch and get married in the US, there'a heap of paperwork involved to get your US marriage certificate to be registered in the Netherlands and have your marital status made official so you're qualified for all the benefits married couples have.
In some countries, as Annoyed points out, you might need to be married in order to share a hotel room. In such situations, it would depend on the laws of the countries involved.
Extreme situation might be a homosexual couple who're in their home country legally married traveling to a country where homosexual marriage is not recognised and only married people may share a hotel room. In that country it's quite possible their marriage would never be recognised, in other countries there might be international treaties in place that handle such a situation.
Best to ask a consulate or embassy if you want to make sure, it's the only way to get the information about your specific information.
From the broad travels I've been at, I've never seen my parents hand anyone a piece of paper to indicate marital status. Apparently the surnames on your passports being the same is for most countries enough to indicate that you're indeed family.