1) Does Limbic Neural Enhancing nanoware affect assensing?

2) Can Adepts counterspell? What if they can astrally perceive?

1) I would say yes, but remember that they only work when there are not significant distractions, so they would not work in combat, etc.

2) Only if they are mystic adepts who have allocated at least one point towards mage abilities. Astral perception would not make a difference.

Assensing is indeed linked to Intuition. Check the table on 4A 120 for a list of which skills are linked to what attribut.