What is the ruling on playing the Takbirat in stores during the first ten days of Dhul-Hijjah, and is that considered an innovation (bid'ah)? And is guiding students to say the Takbir during these days considered an innovation?

Raising recorded voices with Eid takbirat (exclamations of 'Allah is the Greatest') is permissible, as it serves to remind people, provided it does not cause harm, disturbance, or exaggeration. This is not an innovation (bid'ah).

Takbir is legislated during these days, and encouraging it is permissible at any suitable time, as long as it does not lead to neglecting an obligation. It is not restricted to a specific time. So, if a teacher performs takbir for his students to emulate him, or instructs them to do so, it is permissible, but without the students performing it collectively in a single voice.