If you have access to your super and are allowed to withdraw a lump sum, then yes you can gift whatever you like after that. Once it's withdrawn, it's your own cash to do whatever you like with.
There are no gifting taxes. The only impact on you would be if you receive a Centrelink benefit, so check that first.
You won't be able to transfer from your super account to your mother's super account.
If you want to give your mother some super in the event that you die, you need to direct your super to be paid into your estate, and leave your mother a gift in your will.
Your superannuation is preserved for your retirement. You need to meet a condition of release to be able to access your superannuation. There may be taxation implications if you withdraw funds from super - you would need to obtain personal tax advice if you are eligible.
You are not able to 'transfer' superannution to someone else - this is because your superannuation is for your use in retirement. You would have to withdraw superannuation and then the other person would need to contribute to their superannuation. Again, there would be tax implications with this.
As Jacqui has said, you could leave your superannuation to your mother in the event that you pass away, provided you have no other relatives that would superced her as 'dependents' either under superannuation or taxation legislation. You could do a non-binding nomination to your mother if this is the case, however the Trustee of the super fund would have the final say. There may also be taxation consequences, and potential effect on Centrelink assets and Income test if your mother receives Centrelink payments. You may be better to leave your superannuation to your estate, but then you need to make sure you have a Will to ensure it ends up where you would like, and to provide less stress for any loved ones dealing with your estate.
This is general advice only - my only advice is to obtain professional legal and taxation advice from qualified professionals.