Is it possible to connect EXP pedal and AUX switch (with three buttons) at the same time? Or should I look for a MIDI controler instead of AUX switch? I really need the instant load for presets (one for up, one for down – and the third one for bypass) and as far as I've read about it, it can only be done with external switches. And I also need an expression pedal. So, does these two go together or can I only use pedal and one button switch?

You can only use an EXP pedal with a single button AUX switch.  So, you'll want to look for a MIDI controller instead of an AUX switch.