It is very important for you and others reading this forum to understand that the vast majority of people commenting on this forum are in no position whatsoever to articulate practices across dioceses and their propriety.
Most of the people commenting lack any education at the graduate level in the subjects for which they are attempting to offer commentary. Thus they give wrong answers, while trying to effect that they are experts in the field. They are anything but that.
The first questions you should ask of posters here are:
- Are you a priest or deacon?
- Are you a theologian, with a doctorate in your theological specialisation?
- Are you a canonist?
As to your specific points:
- Yes, there most assuredly are parishes which elect not to do baptisms during Lent.
- Yes, there are parishes which require repetition of the baptismal class – for both parents and baptismal sponsors. One should also expect documentation requirements.
- You should comply with the procedures enacted by your parish priest…reserve a spot in the next class and then reserve a spot in the next scheduled administration of Baptism. You should clarify the requirements for the sponsors now so that does not become an issue.
Any person at any moment can experience sudden death…that remote reality is not, however, the basis for how we, as priests, determine the normal administration of the sacraments. I do not instantly receive someone into the Church because they could be run over by a car before the envisioned moment of reception. On the other hand, I have adjusted the schedule, even radically, for persons who were, actually, terminally ill.
If an infant has an actual diagnosis that is life-threatening, the case is handled differently.
The way in which the canons are actually lived by the Particular Churches varies greatly between the developed world, with its low infant mortality rates, and the developing world. Procedures and also expectations are markedly different between, for example, North America and rural areas in Africa.
What you relate indicates your child could be baptised in, more or less, 6-12 weeks. That is not outside the norm nor is it outside of what canon law envisions…at all.