Do I need dumbbells to use MyWeeklyWorkout?

No, you can use a weighted training bag, like a rucksack, for the resistance exercise. There are video demonstrations of the training bag exercises too.

How heavy should I make my training bag?

As heavy as you like. 10kg is a good place to start.

Do you have any tips on making a good training bag?

Yes, check out the article here.

What weight dumbbells should I use?

That's completely up to you, though 10kg is a good starting point for women and 15kg for men.

Can I use a barbell for the program?

Yes. All of the movements with added resistance can be done with a barbell.

Can I do this program if I'm injured?

Without knowing the injury it's impossible to say, but it's always good to exercise caution when training around an injury. Never train through pain and always follow your physio's advice. That said, this program is low impact so may be good if you are prone to impact related conditions like shin splints.