Would it be possible to have a search facility for all activities. A world map for the activities?

Swainster said:

Would it be possible to have a search facility for all activities. Example search on title or place, etc.

Not available on the map, but you can search activities by various criteria, including, titles (with location names), and also comments, in Garmin Connect Web 

Swainster said:

Could we also have a world map with the places we have completed the activities? So that when we click on them the activity data is shown.

Again, not with the Garmin Connect app, but the app Garmin Explore offers similar functionality.

I recommend submitting your requests to Garmin at Share Ideas | Garmin