Notes and rests

When I try to insert a note, an icon appears above the next bar line and the note is not inserted. Why?

You are in "Editing limited to one bar" mode, and you are trying to insert a note that goes over a bar change. In this mode, notes dropped into a bar cannot modify or shift notes in the following bars.
For example, you cannot, in this mode, drop a note which carries over to the next bar.
To allow this, switch this mode off in the Options menu.

How can I hear the notes I drop into the score?

Go to Configuration>Global Setup and check the option "Play note while editing"

Some notes are displayed too close together, but I do not want to resize the whole bar. Can they be moved away from each other?

Disconnect "automatic spacing" (Options menu).
Then click on the note with the "Alt" key held down: you can now move the note, from -100% to +100% of its regular position.
You can also double-click on the note and edit the numerical value of its displacement.
Other solution: if you own Harmony Assistant, activate the Engraver mode: "Score > Engraver mode"

How do I manage chords with different note lengths ?

In this case, the chord length is considered equal to its shortest note.

How to play complex rhythmic patterns, including several note lengths played at the same time on the staff ?

If different note duration are played in the same chord, then your staff probably include several "voices" (left and right hand on a piano, Alto and Soprano voices...). In this case, it is recommended to use multi-voice staves.

You are invited to view the "Merged staves (multi-voice staves) presentation)" ("Windows>Tutorials" menu option in the program.