Which filters are the most commonly used and easier to implement for ECG noise removal?
Different filters are used for different purposes. Removing powerline noise, which usually is concentrated at a single frequency, is usually performed with a notch filter. Removing baseline wandering is usually done with a DC blocking filter, which is a very narrow high pass filter, and is essentially another notch filter with 0 Hz center frequency. Removing EMG is usually more difficult, but you can try doing it with a low pass filter, since EMG has a lot of content in high frequencies.

In order to process EEG signals for simple things like heartbeat rate detection, look at the Pan-Tompkinks algorithm. 

Questions 2 and 3 require some effort on your part, since they're very broad. I've used this book in a graduate class on the topic, and I found it to be very insightful.