I mean something like this:
I'm guessing it means they're played together but I'm not sure.
Also, does the sharp apply to both pitches or just the G?
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Noteheads take up one full space on a staff. (The corollary to this is that noteheads placed on a line take up half of the space on either side of the line.)
When notes are at least a third apart (as in the first measure of the example below), there is plenty of space for two noteheads of standard size to fit on top of each other.
But when notes are less than a third apart (as in the second measure of the above example), there's not enough space to notate the pitches directly on top of each other; doing so just confuses the notation.
So we settle on the solution shown in the final measure, where the noteheads are offset slightly. But in all cases, this notation indicates that the given pitches occur at the same time.
Lastly, accidentals only apply to the staff line or staff space on which they are placed. Since the ♯ here is on the second staff line from the bottom (G, assuming this is treble clef), it only applies to that G, not to the F. If the composer wanted an F♯ in addition to the G♯, s/he would have indicated a second ♯ in the score.
Here are two different screen shots of the same bit of music, from the beginning of the coda in Chopin's second Ballade. You can see clearly that the passage has groups of doubled 16th notes. If you look at the last 16th note of the first barred group in the second measure (a D with a natural mark and an E) it will probably be obvious from the context that these notes are to be played together.
I showed you this example because it has an interesting additional problem of notation. If you look at the fourth and fifth 16th notes in the last group of barred 16th notes in the second measure, you will see that a simultaneous D natural and D sharp need to be played. Here we go one step further with the problem: two different notes that are on the same line or space of the staff and need to be played together. The problem is that you can't fit both notes on a single stem as you can two notes right next to each other. There are different ways to handle this.
This example (which is more common, and is the way that Chopin notates it in his manuscript) has two separate bars on the double notes. You'll notice that this isn't done with any other group of 16th notes, so the only reason that it's done is to accommodate the D/D# notes.
This example forks a single stem in two so the notes can be written next to one another and still be understood to be played simultaneously.
There is no difference in the way these are to be played. I personally prefer the second way of writing this, even though it is less common, because I find the intention more immediately obvious. You'll notice that the two D natural/E double notes before it look exactly the same, whereas they look a bit different in the first example. So all in all, I find the second example a bit easier to read.
Yes, play them together. They are only placed slightly next to each other, and not one above the other, because they are adjacent pitches and would overlap slightly otherwise. The # is just for the G. So the pair of notes are F and G#. (Unless of course the tied note is an F# anyway...)
On fretted instruments (like guitar or mandolin), it is possible to play a unison, such as (for mandolin) an open E string and the B string fretted at 7 to make another E; both are played simultaneously. This is fairly common in Bluegrass.
The Notation for this sometimes shows two of the same notes immediately adjacent to each other and on the same line (or space).