(1) Can a business be transferred under TUPE when the transferor was trading unlawfully?
Yes. Although the undertaking ought to be a lawful one in the sense that if it is for an inherently unlawful purpose "such as a drug dealing or money laundering enterprise" then TUPE would not be applicable, TUPE does not require that all of the activities being transferred be entirely lawful.
The business in this case was a solicitors' firm. It is perfectly lawful to provide legal advice. It may be that the lack of certain qualifications held by certain people may have made some of the firm's activities unlawful at certain times, but the business was not carried out for an unlawful purpose. TUPE did apply.
(2) Where an employment tribunal finds that a dubious "sick note [is] produced simply to add spurious weight to a weak and very late application for an adjournment", must the employment tribunal direct further enquiries to substantiate the sick note before refusing the adjournment?
No. In this "perhaps rare" case, the circumstances did not require the employment tribunal to take any further steps before refusing the adjournment. There was no reason to decide that the attendance of the First Respondent was needed for a fair trial in these particular circumstances (listed in paragraphs 25-26 of the EAT judgment). The decision to proceed was within the employment tribunal's discretion.