Replacement Battery for Dell XPS 15 L502X Inquiry

Good day!

My windows is telling me to replace battery for XPS 15 L502X 2011, Service Tag: <ADMIN NOTE: Service tag removed per privacy policy>

I am willing to buy the original battery, but I have some questions.

1) Will this battery fit into my notebook? Or there is a better option?

2) Do I need a different adapter for this battery? If yes, can you please provide link to it?

3) Can you ship these to Latvia? 

Hope the support can answer my inquiry, as my poor Nora (notebook name :) Is getting pretty hot, and I want to live a long and happy life!

Thank you in advance and Regards,

Alex.