I have the HX90V, so not the exact same camera. But according to the Sony website, this should be the right battery: https://www.sony.com/electronics/cyber-shot-compact-cameras-batteries-chargers/np-bx1#related_products_dynamic_default It is also listed as an accessory of the HX99: https://www.sony.com/electronics/cyber-shot-compact-cameras/dsc-hx99#related_products_dynamic_default
Yes, the following batteries can be charged in the Sony BC-TRX, Cyber-shot Battery Charger: NP-BX1, NP-BN1, NP-BN, NP-FG1/BG1, NP-FD1/BD1/FT1, NP-FR1 and NP-BK1. Or, in short, for all X-/N-/G-/K-/D-/T- and R-series batteries.
Enclosed is the charger incl. battery: Sony BC-TRX, Cyber-shot battery charger incl. NP-BX1 battery (charger, battery)
Yes, they are available. Even incl. battery. (Charger without battery only CHF 3 cheaper).
Sony BC-TRX, Cyber-shot battery charger incl. NP-BX1 battery (battery, charger)
Sony BC-TRX, Cyber-shot battery charger (charger)
Yes, it fits all X models! See also <http://www.sony.ch/de/electronics/cyber-shot-kompaktkameras/dsc-hx90v/specifications> at the very bottom of the scope of delivery!