05 ford focus: continuously flashing open door light on my cluster.

Check fuses I notice they shoe instrument cluster fuses on link of fuse diagram below.
https://www.autogenius.info/ford-focus-2004-2010-fuse-box-diagram/

your door might be open. or the switch thinks it is.

Sounds like the contacts in the buttons are not making contact.. you may want to try taking the remote apart and cleaning the contacts and seeing what happens. If the remote is truely dead you may need a new remote

check remote battery if that doesn't work could be a fuse blown, the accuators are bad but if its doing both doors unlikely theyd both go bad at same time however some vehicles will not work on either if one of the 2 is bad that's me assuming you have a 2 door vehicle. same for 4 door but rear may work as front doors wont or all 4 could not work depends on vehicle. If its not accuator you could have a problem witht he alarm brain and for that take it to a stereo shop they do alarms and most places will check it for $25

At the door jam,check the condition of the pin switch. The switch may be cover with a boot.
Compare the condition of this pin switch to the working side.
Clean the general area and re-test.

well bob wilson this is steve balic and i think i know you! anywho just change the instrument cluster and it be just fine! i just had same problem with my mothers 2003 ford focus and that's what it was!! i got a used one for a new one is big bucks unless u gots! u welcome bud

could be a bad earth or ground on the rear light clusters

Hi there is a couple of screws if you lift the hatch(boot) up unscrew them remove the light cluster and take it from there

there are two seperate horns

sounds like you have a problem with your alarm system. likely the ignition switch is the problem.