I lost the physical copy of Destiny, is there a way I can still play it

You did not get ripped of. You got what you paid for.

From what I could tell, I found that in the bundled items it displays:

• PlayStation®4 System (Glacier White)
• DualShock®4 Wireless Controller (Glacier White)
• Destiny Video Game (Physical Disc)
• 30-Day PlayStation Plus Trial
• HDMI Cable
• Power Cable
• Wired Mono Headset
• USB Charging Cable

I could not find any mention of the "Digital Guardian Edition" so they are not entitled to give you the additional expansion pack.

You may have received the disc for the expansion pack version, and the armory code (although expired), you did not buy the Digital Guardian Edition so your not entitled to anything extra.

Yes, you can play Destiny TTK at the same time, because when you download a digital game onto a different console, it adds a "temp" license for that user, as long as the account that has bought Destiny is on the console the game should be fine.

I hope this helped. Good luck Guardian