Oh Boy! You got a few issues.
Yes your system can boot up without the battery. You’ll need to take the battery out and disconnect the charger. Press the power button for a minute to drain the logic. Once thats done plug the MagSafe back in you should be able to boot un now.
'''But!''' You appear to have a second problem which is then getting it the way! Your SSD does not have a workable OS on it as you are seeing the '''Prohibitory symbol'''
[https://support.apple.com/en-us/HT204156|About the screens your Mac displays as it starts up]
Lets try this restart the system this time and press the '''Option (⌥)''' to get to the Startup Manager. Do you see a valid OS present? If you do select it and your system should boot up. If not you’ll need to boot up from either an external bootable drive or leverage Internet Recovery (this time pressing the '''Command (⌘)''' and '''R''' keys) to then create a bootable external drive so you can then try to fix your internal drive.
[https://support.apple.com/en-us/HT201255|Mac startup key combinations]