How do diagnose network problems
The key to your diagnostics is this statement:


  I think my access point is at fault, as this also happens to other
  devices.


When you have multiple devices exhibiting the same behavior, the best course of action is look for the common thread - in this case your WAP (Wireless Access Point)

You never mentioned what wireless router you are using, but it's probably a safe bet is a consumer grade unit.  I experienced the same issues until I finally ditched that gear and went for more commercial grade equipment.

In your case, even if you didn't want to replace the router itself, you could disable the WiFi functionality and add in "proper" WAPs as described in the linked answer.