Hi.
I 'm a little confused on the part of conspiracy. Do you need to spend 1 Time to move the pointer to the next stack detective puzzle ?

No, you can't spend time to do this. When you investigate a lead (i.e., interact with a suspect or piece of evidence on the map) you have three options:

1: Place an evidence marker on a suspect sheet

2: Place a consipracy puzzle piece of the type your marker is pointing at (then move your marker to the lowest type available)

3: Move your marker to the next highest conspiracy puzzle piece type available.

Hope that helps

EDIT: The rule, if you want it, is on p.25 of the rulebook, under "digging deeper".