in normal situations - you are fired without bankruptcy - then you are still considered as employed in the notice period. So in normal situations your last day of working will be when the notice period expires.
However your case is a special situation, and it is a little out of the scope for our website to answer questions about payments from Lønmodtagernes Garantifond.
I can only guide you about how the rules are in Denmark. If your home-country have different rules/practices, then you must try to contact the relevant authorities in your home-country and ask them.
If you were continue to live in Denmark, your A-kasse are right, when they are telling you that you are regarded as unemployed on the first day of the notice period/"fritstilling". It is important that you on the first day your are "fritstillet" register as unemployed on Jobnet.dk and regardless of whether you apply for payment from LG you still have to apply for job/search jobs like if you were unemployed.
Its a requirement for being eligible to get payments from LG, that you are actively searching for jobs in the notice period.
If you were continue to live in Denmark, and because it will take some time before LG can pay out money to you (the court have to confirm the bankruptcy), you would be able to get dagpenge in the notice period (as a loan from the A-kasse). When you get the money from LG, they will have to be paid to the A-kasse to pay out the loan. Off course its only possible to get dagpenge if you meet the requirements for being eligible for dagpenge.
I will recommend you to contact Lønmodtagernes Garantifond and get this confirmed. You can find more information and contact-info here: https://lifeindenmark.borger.dk/Living-in-Denmark/Work/Compensation-if-employer-is-bankrupt.