This is the beginning of Mendelssohn's Song Without Words, Op. 19, No.1. Are the orange-circled parts another instrument or voice? What about the bold red-circled parts---how to tell if they belong to that "other instrument/voice" or the bass notes? Did I miss any? (Previous similar threads don't fully answer my question, as I'm really interested in this particular piece)
They belong to "another voice" in a metaphorical sense. You still play them on the piano, but you want them to sound as an independent voice from the other "voices". When multiple voices are present on the same staff, the stem directions tell you which voice is which. It is possible for an inner voice to move from one hand to the other (i.e., one staff to the other), but this is typically either indicated by an explicit notation or obvious from context. This is not an issue in the Mendelssohn Op. 19 No. 1.
Overall, the notation is typical of how polyphony is written for piano. Here's an excerpt from a Bach fugue in three "voices" (Fugue in C#, BWV 848):
And here's a recording by Sviatoslav Richter where you can hear how the "voices" are articulated.
The only one you missed is the whole note in measure 10.