Learn Russian: Он всегда может справиться и найти всякие полезные вещи. - He can always cope and find all sorts of useful things.

The lesson focuses on the difference between these universal pronouns. Всякие here translates to 'all sorts of' or 'various kinds of'. If we used любые, it would mean 'any useful things at all' (as in, whichever ones you choose). In this context, the speaker is emphasizing a diverse variety of found objects rather than absolute freedom of choice.

The sentence includes всегда (always). Why are the infinitives справиться and найти in the perfective aspect?

While всегда usually triggers the imperfective aspect for repeated actions, here it modifies the verb может (is able to). The infinitives are perfective because they describe the successful result of his ability on any given occasion. He is always able to 'successfully cope' and 'successfully find' things.

Can you use справиться on its own without saying what he is coping with?

Yes. Although справиться is typically followed by с plus the instrumental case (like справиться с задачей for 'to cope with a task'), it can be used without an object to mean 'to manage', 'to get by', or 'to figure it out' in a general sense.

What case is всякие полезные вещи in?

It is in the accusative case because it is the direct object of the action найти (to find). For plural inanimate nouns like вещи, as well as their modifiers, the accusative form is identical to the nominative form.