There are a number of places in New York—for example, Manhattan, Park Slope, and northwestern Queens—where the streets running in both directions are numbered. What’s the convention for which number is given first when you’re naming an intersection? For example, is “Third and Fifth” 3rd Street and 5th Avenue or is it 3rd Avenue and 5th Street? Is there a different convention in each of the places where there are intersections like this?
Honestly? There's only a few dozen intersections where this sort of ambiguity is a thing; while Street, Avenue is the most common usage, I've heard the reverse often enough, as have most New Yorkers, that if you're talking about an intersection of two single digit roads, you're going to qualify/classify it.
There are a variety of reasons that this doesn't matter much elsewhere in New York.
In Manhattan, it's because Avenues only go up to 12, and there are some 200 crosstown streets; the West Village and it's peculiar bending of the grid and named streets means that you have very few intersections of numbered Streets with numbered Avenues south of 14th. (Specifically, you have two 'First and Seconds', and a pair of "Third and Sixths". That's it. You can confirm that on a map if you like.)
In Brooklyn, you have several separate street grids. Numbered streets intersect with named streets in Williamsburg and Greenpoint, but be sure to remember the North or South designation. Over in Park Slope, you run into the only real potential source of confusion - but they're all within a fairly small 5x5 block area - it's not a huge issue. In South Brooklyn, the intersecting streets are numbers and letters, so again, not a problem.
Finally we have Queens. Queens is, to the uninitiated, a total mess. The thing to understand about Queens, is that you can have as many as 5 streets with the same number running in parallel to each other, and another 5 running perpendicular across the entire borough. In Queens, you'll never give a location as number and number, because the suffix (Street, Avenue, Road, Drive, Court, Place, Terrace, Crescent, etc.) is essential information in the address.
To use your own example, there is literally no such place as "Third and Fifth" anywhere in NYC anyway; and most other, similar examples you might contrive don't actually exist to cause the confusion you'd imagine that they could. It generally doesn't matter, because for the vast majority of cases, the numbers themselves imply where the pair meets.
In Brooklyn for example, addresses are designated as follows:
On the other hand, the corresponding location on 13th Avenue would be:
If so, either way would be appropriate, depending on your point of reference: If you're talking about a place on 13th Avenue, you'd say 50th and 13th, but if you're on 50th Street, you'd say 13th and 50th.
Some areas in other parts of NYC have similar addressing schemes - when possible, I'd say let the address be your guide. Unfortunately, as other answers have noted, in certain parts of NYC, particularly Queens, all bets are off...
Avenues run uptown and downtown, streets run crosstown. At least that is how it is in Manhattan. So if you were downtown at lets say 1st Ave and 1st street, you can take 1st ave all the way uptown to 100th street. (it goes further than that, but I think you get the point)
In Brooklyn you have the same kind of setup. Main avenues running down the streets. So it would go something like E24th st & Avenue J.
Generally the lingo for NYC streets is dependent on where you are. In my mind, I have always gone street-ave. So I would interpret that as 3rd street and 5th ave, but I can easily see someone like a cab driver getting that messed up if you say third and fifth. Whenever I get in a cab I always give the more specific street first, followed by the intersection. So if I was going to Grand Central, I would say (normally grand central but for these purposes) 42nd & Park.
Think about an address, the more specific part comes first, and then the broader location. So you say Albany (specific), New York (non-specific). Apply the same concept to streets. There are only a handful of avenues but there are a lot of streets.