1) Yes, you can sign up now. Will be free of cost until graduation. But if you sign up now, you must remember to contact your A-kasse when you graduate, and tell you want to have graduate-rights.
2) The suspension can only be settled during periods when you are entitled to benefits from the A-kasse. This means that you are registered at the job center (are available for the labour market) and submit a unemployment benefit card.
So you can not settle suspension period, if you work. Because if you work you are not available for the labour market.
3) With the jobtype you have where you can leave without notice, then it will be possible for you to get supplementary benefits, but only for a period of maximum 30 weeks. You can read more about supplementary benefits in these Q&A:
4) When you say vacation in March, I assume your suspension period is over. You will not be able to get benefits for the period where you have holiday. When you go on holiday in another country you are not considered to be available for the labour market (you will not be able to take job with one days notice). You must report any holiday on the montly unemployment benefit card, that you fill out online every month at your A-kasse's website.
The A-kasse will deduct the days with holiday in the amount you receive in Unemployment benefits ("dagpenge"). The situation will be same regardless of you search job or not while on holiday.
If you instead would be able to go on holiday in the suspension period, then it is possible as a newly graduated to do that, if you go to your homecountry.
Remember to report vacation to Jobcenter no later than 14 days before the holiday starts.
Anders Weber, www.a-kasser.dk