Is it possible to combine a HC-05 Bluetooth module and a shrinked arduino ATtiny45/85 to control a motor, LEDs, or other components?
The ATTiny has 5 usable pins (6 if you decide to reuse the Reset pin). Since it doesn't have UART hardware, you need to use a SoftwareSerial library to communicate with the HC-05. This takes 2 pins. To drive a motor, you need at least 1 pin. If you need to control it's speed (instead of just on and off), you would need to generate a PWM signal. Either through hardware timer or by code. (The ATTiny has 2 timers. When you want to use PWM with hardware timer, you need to make sure, that the SoftwareSerial library doesn't want to use the same timer). For the servo it's similar. The servos position is encoded by pulse lengths. (Make sure, that the used library doesn't want to use an already used timer)

Unless you use techniques as multiplexing or matrixes (which are not that useful for a small number of LEDs), you need 1 pin per LED.

Depending on what exactly you want to achieve, you will pretty fast run out of pins. Since you are a beginner you should start with a chip with more pins. If you don't know, how exactly your project will be in the end, better by a chip with maybe double the pins, that you think you need. Develop, until you really know, what functionality your project should have, what not and what you need to achieve that. Based on this you can choose the correct chip, that is as small as possible. Also consider using SMD parts in the finished version. There are mostly much smaller, but also more difficult to solder.