Powering Arduino pro mini v5 with NimH batteries
Well, NiMh aren't best choice for such device, especially when it comes to charging - they require full cycles of charging - and that can be problematic for users of your application. Moreover they degrade with time much faster than other types of battery.

I would suggest using Li-ion battery. Nice and cheap source for them are batteries from cellphones - you can buy such batteries (like from Nokia) very cheap (they probably aren't original, but it doesn't matter).
They require special charging but you can easily solve that buying - also very cheap- Li-ion chargers as standalone modules which even have built-in usb connector - so you don't need to provide special charger to your product - users can uses their phone charger or PC.

If you really need small size - you can use LiPo batteries - they are usually used in models (like drones etc), but they are really small size.

Calculations seems ok. But for better results you need to make tests.

And you don't necessary need 5V - you have to check datasheet of your componets. AVR on arduino usually can work with lower voltage. Transmitter and sensor - I don't know. Look into respective datasheets/manuals it should be even written on the first page.