As with all leases, your rights and obligations are spelled out in the lease or are otherwise determined by statute. Generally speaking everybody has access to common areas, that's what makes them common. Often there are limits or retrictions on what tenants can leave or store in common areas. Check your lease. If it says you can't leave or store things in common areas then yes your landlord can move them. Depending on the lease the landlord may even be allowed to dispose of the items. Again, check the lease. If it does not address the issue then I encourage you to contact a local attorney for further clarification.
With regard to your second issue, there's no question there. My suggestion is that you let her know that the recycling and trash are not yours. Be sure to make all communications in writting and keep a copy (email is good too) so there is a record of your conversations.