My iPhone cracked and isn't working properly. How can I get it fixed or transfer the info to a new phone without going to a store?

If you have a backup to either iCloud or a computer you can use the iCloud or iTunes/Finder methods described in Transfer data from your previous iOS device to your new iPhone, iPad, or iPod touch - Apple Support

Also, any data synced to iCloud via the options at Settings > [username] > iCloud > (e.g., iCloud Photos, Contacts, Notes, etc) will automatically sync to a new phone (using the same Apple ID) when those settings are turned on in the new phone.

What can you do? Does it work at all? Maybe there's intermittent operation or random actions like inputs happening without anyone touching the screen? Or is it broken? Just the touchscreen damaged?

It could be tricky if Apple would "repair" the device. Often that's done as a complete replacement in lieu of actually fixing your device and returning it to you. Apple's terms state that it's the responsibility of the customer to perform regular backups and that any service by Apple could include complete erasure of all data on the device.