1. Yes, but probably not the way you think they do. Airlines either have ab-initio cadet programs, (through which a person with no licenses at all goes on to usually obtain a MPL in order to work for the company), or they have "NTR Cadet" job openings, which means that they are looking for people with a fATPL. If you want to go in as a "NTR Cadet", you'll have to get the fATPL on your own. You can get a fATPL on your own, but for a MPL you need an airline to support you.
When getting a fATPL, you always get your licenses individually (look at them as four big chunks: PPL, ATPL theory, CPL/IR/ME, MCC/JOC). The difference between modular and integrated is that, when going modular, you can take a pause between the licenses. When going integrated, you will do them all one after another. The modular path is thus more flexible. Modular also means that you can get a PPL from school X and the CPL/ME/IR from school Y, while integrated means you get them all from one school.
2. Most cadet programs aren't cheaper. There are some programs which are easier to pay, but not cheaper. Take the Wizz one as an example. You only have to pay 18/22k upfront, but they will deduct some 40/50k from your salary while working for them. Your total will be around 60/70k, which is more expensive than the modular route, but it is more affordable because you only have to pay 18k upfront. Job security is indeed a great advantage and sometimes it is worth the cost difference, but this is subjective. The statement is true. You will usually have to work a few years for the company. Leaving earlier means you have to pay the costs of your training.
4. They will not fine you. It is not a fine, they will just ask you to pay for your training.
After your training, they will give you a job and they will deduct the training costs from your salary. If you left earlier than previously agreed, they wouldn't have had enough time to deduct the whole amount. Therefore, you will be required to pay the difference.