Piano Forum

A trill is: (re) do re do re do.
A mordent is: do ti do.

In measures 5 and 6, it's a mordent.

In measure 8 (technically, it's measure 7; you don't count the pickup measure) it's an extended trill.  If you look at the last two notes of the measure, you will see that it is a termination of the trill.  In other words, it's a trill with a written out mordent: do re do re do ti do.

In measure 16 (measure 15) it's a short trill of 4 notes: do ti do ti.  You may wish to start the trill on the principle note if you prefer the aural aesthetics better: ti do ti do ti.

By "inverted mordent", you are referring to a trill.  (It's incorrect and gave me a hell of a time figuring out if you were referring to a trill or a mordent.)  Whether or not to start on the principle note or the auxiliary note depends mostly on musical preference.  However, in this case, start on the auxiliary note, the note above, in all instances.

... how would you play the second measure of the first end section (start a trill on a note that has just previously been played in 8th note)? Would you quickly switch finger or play with the same finger?

Yes, you would repeat the D: D | D C# D C#.  It does not matter if you use the same finger to repeat the D.  In my preference, I would switch from a 2 to a 1 because it's just a bit easier on my muscles to trill with a 1-2 from a white to an adjacent sharp than a 2-3.

I hope this helps.