Learn Russian: Врач порекомендовал моему отцу отдыхать.

The verb порекомендовать (to recommend or advise) requires the Dative case for the person receiving the advice. This fits with the rule that verbs of assisting, advising, or giving take the Dative without a preposition. Think of it as giving a recommendation to someone.

Where did the 'е' in отец go when it became отцу?

The noun отец (father) has a 'fleeting vowel.' This means the vowel е drops out when you add case endings. Therefore, the Dative form is отцу, rather than отецу. You will see this happen with other cases too, like the Genitive/Accusative form отца.

Why are the two verbs in different aspects (порекомендовал is perfective, but отдыхать is imperfective)?

The doctor giving the advice was a single, completed action in the past, so we use the perfective порекомендовал. However, the rest itself (отдыхать) is general, ongoing lifestyle advice ('take it easy in general'), which requires the imperfective aspect.

Could we use чтобы (so that/in order to) here instead of just the infinitive отдыхать?

You don't need it! With verbs of advising or ordering (like рекомендовать, советовать, or сказать), you can simply put the person in the Dative case followed directly by the infinitive of the action they are being told to do.