1) Yes, with a completed education that gives you graduate rights, you will be entitled to start to receive unemployment benefits within a period of 2 years. So if you get fired before you have enough income earned from job to fullfill the income-requirement (as a salaried person, you obtain right to benefits if you have earned/have had an income of at least DKK 223,428 during the last 3 years), then you have the option to use - fall back on - your graduate rights.
Be aware that you still have to contact your A-kasse at latest 14 days after finishing education to have you status with the A-kasse registered as graduate. This will give you the graduate rights, so you can make use of them later.
2) The same applies in this scenario - as long as you remember to be registered in the A-kasse with graduate status - you have right to unemployment benefits within a period of 2 years after finishing education. If you get fired after 1 month - you can get benefits according to your graduate rights.
The 3 months period with work as an newly graduate - you talk about, has not effect on your right to get unemployment benefits, BUT after 3 months full-time work you have the right to get your benefit rate recalculated - so the benefit rate will be based on your income in the 3 months and therefor will give you a higher benefits rate compared to the lower graduate benefit rate.
You are welcome to comment on this,