If your model is the Mid 2012 13" MacBook Pro with built-in DVD aka MD101, you should know that they burn through SATA-HDD cable like crazy. When this cable fails, it does exactly what you are describing: a flashing folder.
Another cause for the flashing folder is a bad disk. And and great way to diagnose this and isolate the issue to disk or SATA cable is to boot from USB to confirm working disk and OS. Which you just did.
So now you need to replace the SATA-HDD cable.
Here's a link to the replacement guide. It also has a link to the needed replacement part.
It is an easy job, just don't forget to disconnect the battery first and reconnect it last.
Bonus points: if your old HDD is still working, and you want to transfer your old system and softwares/applications and personal files, then after replacing the internal SATA-HDD cable:
1/ Mount the old HDD inside your computer and pull a time machine back-up to a spare/empty HDD that should have the same or more capacity than your existing data on the old HDD. For example, if your disk is the factory 500GB, and it contains 150GB of data, you need an empty disk with more than 150GB capacity to pull the time machine back-up. If your 500GB HDD is full, you will need an empty 500GB capacity disk. Once done, remove the old HDD and mount the SSHD in the computer, then restart your computer by pressing command+R and following the instructions to restore your time machine back-up.
2/ You might be able to install the SSHD inside your computer, connect your old HDD via USB, start your computer while pressing command+R, and use Disk Utility to restore your old partition to your new disk. Be careful to choose the source disk wisely, otherwise you risk deleting the source/old HDD.