If I’m person from EU and I work for USA remote company. Is it possible for that company to pay taxes, insurance and pension contributions. And if yes how do?
Usually not. I was in the same situation and had to register a company in the EU. That way I pay my own taxes, insurance and pension contributions. If you do that, make sure to calculate the extra cost for insurance/taxes and raise your rates accordingly.
What was cost of the registration and is it good option for grads?
Think that depends on which country you’re in as well as the type of company you’re registering.
You don't need to register a company unless you need to employ people yourself to get the work done. You can just register as self-employed. The details will vary depending on where you are. You'll most likely have to convince the tax authorities that it is genuine self-employment and not your employer dodging taxes. If you have an actual job with an employer abroad, you'll need to ask your local tax authority how to handle it.
Your hourly rate when self-employed should be around 2.5x the equivalent salaried hourly rate. This is because the employer is not paying for your admin time, training, equipment, office costs, holiday pay*, sick pay*, or contributing to your pension, or doing your taxes for you*. And because you don't have the security of a salary and will need to spend a lot of time bidding for work that might not pan out, while they get the benefit of having you on tap with no long-term commitment.
*these are not USian things but as a European you will still be losing them
It depends a bit on the field you work in but self-employment is not generally a great idea for a fresh graduate. Your degree qualifies you to become qualified and you can't easily do that on your own. Lack of experience will also make it hard to get the kind of work you need to develop your skills and get better work.
There would be nothing legally requiring a USA company to do any of that. When you negotiate the contract you should either increase your base price to include that or explicitly add it to the contract. It doesn't hurt to pay a USA contract lawyer to write/review the contract to make sure it covers what you want.