High Sierra installed but boots into install partition

Hi gersa,

For the first issue, try the following command using Terminal.

sudo rm -rfv /Library/Caches/com.apple.iconservices.store; sudo find /private/var/folders/ \( -name com.apple.dock.iconcache -or -name com.apple.iconservices \) -exec rm -rfv {} \; ; sleep 3; killall Dock; killall Finder

If the issue persists, it could be permissions related. See the following article for guidance.

Resolve issues caused by changing the permissions of items in your home folder

https://support.apple.com/HT203538

For the second issue, change the startup disk. Then, use Recovery mode to remove the unnecessary partition and reinstall macOS.

All the best.

In addition to Sean's advice: go to System Preferences->Startup Disk and select your SSD as the startup disk. Then see if upon restart it will "take".