Does the MKR WIFI 1010 support the arduino mouse and keyboard controller?
Yes, the Arduino SAMD boards are supported by the Arduino Keyboard and Mouse library.

Nano 33 IoT is a smaler alternative to MKR 1010.

But the MKR1010 or the Nano33 IoT with original NINA firmware doesn't yet support classic Bluetooth like the HC-05 module. The ArduinoBLE library and the NINA firmware support only BLE which is different. 

But there is a simple way to replace the NINA firmware with the ESP32 Arduino basic Bluetooth sketch SerialToSerialBT and then use it from the SAMD. This way the NINA module will be degraded to something like a HC-05.