Learn Russian: Стол занят, мне некуда положить ноутбук. - The table is occupied, I have nowhere to put the laptop.

негде means 'nowhere' in terms of location (answering 'where at?'). некуда means 'nowhere' in terms of direction or motion (answering 'where to?'). Because placing the laptop involves moving it to a destination, we use the directional word некуда.

Why is the sentence built with мне instead of я?

Words like некуда, негде, and некогда trigger an impersonal structure. The person who has 'nowhere to go' or 'no time' is the receiver of that situation, so they are put in the Dative case (мне). It literally feels like 'to me, there is nowhere to put...'

Why is the verb положить in the perfective aspect here?

You are referring to a single, specific result you want to achieve right now (getting the laptop onto the table). When talking about a specific, completed action like this, Russian uses the perfective infinitive.

What case is ноутбук in? It looks like the subject.

It is actually in the Accusative case, as it is the direct object of the verb положить (the thing being put). Because ноутбук is an inanimate masculine noun, its Accusative form is identical to its Nominative form.

Why does the sentence use instead of the full adjective ?