Mixolydian Mode

2nd position and cross harp are the same thing. Mixolydian mode is not the same thing as 2nd position. However, 2nd position IS the most common way of playing a mixolydian tune.
2nd position is playing in the key a 5th above the home key. For instance playing in G on a C harmonica. On a C harp 1st position is key of C and the key notes are 4 and 7 (also 1 and 10 are Cs). On the same C harp 2nd position is key of G and the key notes are -2 and 6 (9 is also a G). In first position the seventh tone is major. In the key of C this would be a B. In the key of G the major seventh is an F#. In mixolydian mode the seventh is flatted. In the key of G the flat seventh is an F. On a G harp all the Fs are sharp. The easiest way to play in G mixolydian is to play G on a C harp. Draw 5 is an F note on the C harp. However, there are other ways to play mixolydian. You could use overblows/overdraws or bends to manufacture notes that are not on your diatonic. Or you could use a chromatic harmonica and play mixolydian in any key. In the key of C the seventh is B. The flat 7th is Bb. You could play your C harp in first position in C mixolydian mode if you could somehow come up with a Bb note. Easy to do on a chromatic.

2nd position is not automatically mixolydian. You can play major key melodies in 2nd. For instance the songs Amazing Grace and Precious Memories do not use the 7th degree of the major scale at all. So you can play them in 2nd position on a standard Richter and not worry about the flat 7th. The flat seventh is the draw 5 in second position. Also you can play a tune in second and use the major seventh as an overblow. Another way to play major key melody in 2nd is to use a harmonca with the draw 5 raised half a step. On a C harp the F is -5. If you raise -5 half a step to F# you can play the major seventh. This is called Country Tuning and is used by Charlie McCoy, Buddy Greene and others to play major key melody in 2nd position. Other alternate tunings raise the -5. Melody Maker, Major Cross, and Circular Tuning all raise the draw 5 so you can play the major seventh in 2nd.

I sometimes play in mixolydian mode on the Melody Maker harps. To do this I play the Melody Maker in 3rd position.

Me too, Sarge. I can’t get special harmonica tuned harps to play certain modes. I see the music as written, determine the high and low notes and select an instrument I can play them on. Sometimes i’ll Change the music key or play lower octaves or higher ones, whatever sounds pleasing to my ear. Perhaps sometimes i’m Playing in different positions or modes and don’t know it.

Old Joe Clark is mixolydian. When I play it on a G harp I'm playing in the key of D. Not all fiddle tunes are mixolydian. Most are not.
Bending draw 3 is very common because it provides notes that are missing on the Richter harp.
Draw 7 shouldn't bend much. Draw 7 is B and blow is C which is higher. Normally you bend the higher note down toward the lower note. There isn't a note between B and C. When the blow note is higher (holes 7 through 10) it is more common to bend the blow note down, not the draw note. In fact, trying to bend notes that don't really bend can put a lot of stress on reeds and lead to reed failure.
This is assuming a standard Richter harp. I saw in your other post that you like the Power Bender tuning. Maybe there is a 7 draw bend there.
Like you, I play a lot of first position on fiddle tunes and old time music. But I also play a lot of second position. It really depends on the tune.

I learned some old fiddle tunes in the 90s and 2000s by going to bluegrass and old time festivals. Yew Piney Mountain is one I heard of but I have never tried to play it. It must not have interested me or perhaps it isn't popular here in the Midwest. It seems to have several versions. I Googled it and found this information that may help you.
thesession.org/tunes/15693
banjonews.com/2018-06/lester_mccumbers_yew_piney_mountain.html
I saw it listed as A Dorian, G major, and E minor, depending on the version. Over the years I have tended to avoid tunes that have many variations unless I know a certain version is popular regionally.