What is the maximum number of sensors that can be operated on a single Aurdino unit
Will the Aurdiuno uno ... buy an Aurdiuno mini?


The main differences between the uno and mini are:


footprint
6 (uno) vs 8 (mini) analog inputs


You will get two extra analog inputs and a much smaller size at the cost of the shield not fitting anymore.


  Now, I would like to run the following mechatronic devices: 3 bi-direction motors, 1 servo, 1 hall effect sensor (digital), 2 infra-red sensors, 1 magnetometer, 1 ultrasonic sensor, Will the Aurdiuno uno paired with the motor sheild be able to run these components?


According to the website the shield supports up to 4 motors or 5 servos, so i would guess that it would work.

if the shield can't do 3 motors and 1 servo at the same time, then you can just use the servo library and a pin for that.

The shield requires 3 pins to function; so you need 3 or 4 pins depending on configurations.

Inputs 1 digital, 4 analog; 1 Digital I\O, 1 Analog In

so it looks like you need 4-5 DIO pins and 4 analog pins, which the uno has so yes.

If you were using motor driver boards instead of a shield the cost would be

2x PWM per Motor/driver combo = 6 PWM
1 DIO for servo
1 DIO for digital input
4x Analog Inputs

Which also fits in the budget all 6 of the PWM DIO pins, + 2 more (leaving 6 of the 14) and 2/3rds of the analog pins (also good you can leave A3,4 alone since they share a timer and only use the other ones)