Jean-Paul Banh, DDS

At 22 years old, I would suggested that you do your best to keep your original teeth. If this means getting crowns, do so. Crowns are also much cheaper than implants. Yes, eventually all crowns will fail, and if at that point, the tooth is no longer restorable, then extraction and implant can be an option. FYI, implants can also fail.

Most likely all you need is a filling. However there's a chance that you might need a root canal if your nerves are infected. If that's the case, you will also need a crown.

Sorry to hear about your situation. The first thing that you need to do is to extract those remaining roots and get rid of your pain and infection. Then you can to decide what you want to do with the missing tooth (implant or bridge).

Invisalign could be an option.  It might require trimming some teeth.  I would recommend that you go see a dentist and get a consultation and simulation.  The simulation will not only show you what the results of the aligners will be but also tell you how many aligners you will need (i.e. it will tell you how long it will take).  

Tooth bonding is always a potential solution; however, based on the dimension that you listed, your central tooth will look square (9mmx8.5mm) if you try to close the gap.  Braces will give you a better aesthetic results, but it will cost more.