Not sure if they have any software incorporated that can do the palm detection.
However, you can use an hand glove (buy one or make a DIY from an old sock) to prevent your palm to be detected as input.
Hope that this can help you somehow.
Solution to the problem 2 : Just install an valid apk for your current android versions, in my case, 4.0.3
Solution to the problem 1 : Download and install (using the browser) a new playstore apk for your android version from another trusted store/site, like 'apkmirror'.
Its Android, so most apps should work by sideloading. The only way to know if the app is incompatible is physically testing the apps on the device. If they turn out not to work you can simply flash a custom rom and problem solved.