Your notice period is just the same as any other period of employment so whether you take sick days before your notice or during it, there is no difference legally
As to the notice period, if there is a written contract in place and it contains a specific notice period clause an employee must give if they wanted to resign, they will be contractually bound by it. Therefore, if the employee fails to honour this notice period they will be acting in breach of contract. Whilst no one can physically force them to work through their notice period, it would instead allow the employer to sue them for compensation for losses/damages resulting from their breach.
In reality such claims are very rarely made. This is mainly due to the costs and time involved, also the relatively small damages that can be recovered. The employer has to show that actual losses have been incurred and often that is not easy to do. The most common damages they would claim for are if they have to engage temporary cover for the employee’s duties and the extra wages they have to pay them or recruitment fees for recruiting a replacement at short notice.
So whilst there is no way of predicting whether the employer will take this any further, chances are that they will not. A more likely outcome is that the employer refuses to provide a reference in the future or if they do, it could mention that the employee had breached their contract.
It is therefore best to try and negotiate a mutually acceptable notice period that would suit both parties. However, if that is not possible and there is a pressing need to leave early, that is still a possibility, subject to the risks identified above.
Does this clarify things a bit more for you?