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In Google Hangouts, if I click the contact bubble (the circle on the right where their picture would be) it brings up their contact card. If you long-click the number there, there's an option to copy it to the clipboard. NOTE: you may need to add the number as a contact to view the number in the "contact card."
If your device has a USB OTG port (My Xperia Z1 Compact and Nexus 7 do, and I believe most recent devices do) you may be in luck - get a USB OTG adapter (mine was from a pound shop - if your in the uk try Poundland) and a USB mouse - plug them in to your phone and you can then control it. Its a bit of a faff at first but better than nothing.
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