Pinball Machine: Is Arduino Mega Capable?

Hello! New to Arduino and the site. I'm in the process of making a homemade pinball machine. I have gathered most of the parts I will need (playfield machinery such as flipper assemblies, stationary targets, lane guides, ramps etc.) and know how to control the solenoid coils (I'll be using a 74HC595 Shift "Output" register, each of its outputs wired to the gate of a mosfet, which will energize the solenoids with high voltage) and know how to make the scoring display (I'll be using a MAX7221 LED driver controlling 7-segment LED displays). Also included will be a switch matrix connected to Shift "Input" registers that the Arduino will read, adding points accordingly and an LED matrix for lighting around the playfield.

But since I've never actually worked with an Arduino before, I'm beginning to wonder if an Arduino will be able to do all of the same time. Maybe my question is: Can the arduino do all of these tasks simultaneously? Or if not, is there a way to cycle through all of these processes (check the switch matrix, run the leds, run the scoring display and control the mosfets/solenoid coils) in a timely fashion? I'm thinking of using the Arduino Mega as it has plenty of I/Os for this project. Is there an alternative or is this a good choice?

Any answers, feedback or advice is very appreciated, thanks!

Can the arduino do all of these tasks simultaneously?

What tasks? Displaying a score and toggling some pins is not a challenge.

Yes, the Arduino can do more than one thing at a time, using millis(). It is a way of programming, but sooner or later most of us will use millis().
This is the most simple example with millis() :

With millis() a number of software timers with a certain interval can be created. For example a software timer to update the display 4 times a second. Another software timer to read the inputs 50 times a second. And so on.

Sometimes I use millis() to create just a single main software timer (like a system 'tick'), and use different counters to do different things. Like these sketches : Analog Sensor Setpoint - #49 by Peter_n - Programming Questions - Arduino Forum

I strongly prefer the Arduino Mega 2560 board. The more simple the hardware, the better it is. There are prototype boards (shields) that you can put on the Mega board and solder wires to the prototype board.