In which time zone do I see my timeline?
All the scheduled messages in the Timeline are shown in your time zone. You can choose/change your time zone in Settings → Organization. Read more
If I choose the option “send next message on the nearest free sending slot”, will the messages go out now, or will it take the timing conditions into account?
The message will go out immediately if the sequence timing requirements can be met and there are free slots in the daily sending limits. Otherwise, the message will be scheduled for the closest date when the requirements can be met. The sequence requirements and sending limits are most important.