When sending Mishloach Manos through a Shliach:
Do I need to hand the Mishloach Manos package to the Shliach or can I ask the Shliach to pick up something from the store for me and the like?
The Shliach can pick it up from the store.
When does the Shliach need to be appointed as such: can it be done before Purim?
Yes, but he must deliver the Mishloach Manos on Purim day.
ראה ערוך השולחן סימן תרצ”ה סעיף ט”ז.
Can the Shliach prepare the Mishloach Manos before Purim and just do the delivery on Purim?
Yes, as above.
Note: The above is only regarding fulfilling the actual Mitzvah of Mishloach Manos. However, if you are giving Mishloach Manos to (at least) one person in a regular way, all the restrictions above do not apply while giving to others.