k7300

Do you have to use 15V DC 800mA adapter or you can use 9V or 12V?

Just make sure kit supply exceeds fully charged battery voltage with say 3V.

so it says that with this kit you can charge batteries with 1.2V-9.6V. So I can take some adapter with 12V? And what about mA? 800 or less?

Voltage: Most likely a 9.6V will not correctly charge with 12V.
An unregulated 12VDC charger will probably to the job.

Current:
Once again this depends on the charge current you wish to use.
Adapter must be able to supply at least the required charge current.