[Baertram]

I think you need to put it into the callback function of the event_player_activated so it gets called everytime a zonechage/reloadui takes place:

local function Callback_Player_Activated(...)
--remove borders at action buttons
for slotNum = 3, 9 do
        local button = ZO_ActionBar_GetButton(slotNum).button
        button:SetNormalTexture("")
        button:SetPressedTexture("")
        button:SetDisabledTexture("")
        end
end
 
--Put this somewhere where your addon gets initialized, e.g. inside callback function of EVENT_ADDON_ON_LOAD or just at the bottom of your addon    
--Register for the zone change/player ready event
    EVENT_MANAGER:RegisterForEvent("MyAddonName", EVENT_PLAYER_ACTIVATED, Callback_Player_Activated)

Like written in the code the addon should load some function as it loads up, by event EVENT_ADDON_ON_LOAD. Replace MyAddonName with the name of your addon, like the lua filename is.

local function Addon_Loaded(eventCode, addOnName)
--Is this addon found? Put your addon's name here!!!
    if(addOnName ~= "MyAddonName") then
        return
    end
    --Unregister this event again so it isn't fired again after this addon has beend reckognized
    EVENT_MANAGER:UnregisterForEvent("MyAddonName", EVENT_ADD_ON_LOADED)
 
    --Register for the zone change/player ready event
    EVENT_MANAGER:RegisterForEvent("MyAddonName", EVENT_PLAYER_ACTIVATED, Callback_Player_Activated)
 
end

At the bottom of your addon's lua file put this then to register the event on addon load callback function:

EVENT_MANAGER:RegisterForEvent("MyAddonName", EVENT_ADD_ON_LOADED, Addon_Loaded)

About the weapon bar change:
I do not know how to check on this but I'll have a look.

EDIT:

There is an event for this too. So put this into the ON ADDOn LOAD event callback function:

EVENT_MANAGER:RegisterForEvent("YourAddonName", EVENT_ACTIVE_WEAPON_PAIR_CHANGED, WeaponSwapped)

And this somewhere above:

local function WeaponSwapped(...)
--remove borders at action buttons
for slotNum = 3, 9 do
        local button = ZO_ActionBar_GetButton(slotNum).button
        button:SetNormalTexture("")
        button:SetPressedTexture("")
        button:SetDisabledTexture("")
        end
end
end

As the code is duplicate now try to put it in a nother local function and then call the function inside the Weapon swapped callback function AND the player activated function instead.