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To check the phone speaker, you'd need to dismantle that. You could just test it by connecting a battery to the terminals for it and seeing if it makes a noise.
For the button, you'd need to open up the buzzer box. I assume there are screws under the things circled in red. It's perfectly plausible that the button for your flat is the only one not working and that it's still the button box. Maybe the weather seals have worn out.
I haven't but I would check two things:
I would assume that the button downstairs has gotten dust / grime in it over time and the connectors aren't working any more. Some rubbing alcohol should solve it.