Learn Russian: Вы можете поговорить с любым переводчиком о любой проблеме. - You can talk with any translator about any problem.

The pronoun любой declines just like a standard adjective to match its noun. In the first half, it follows с (with), which requires the instrumental case, so it becomes the masculine instrumental любым to match переводчиком. In the second half, it follows о (about), requiring the prepositional case, so it becomes the feminine prepositional любой to match проблеме. Coincidentally, the feminine prepositional form looks exactly like the dictionary form!

How would the meaning change if we swapped любым for каждым (every) and любой for каждой?

Using forms of каждый changes the focus from 'free choice' to 'entirety'. С любым переводчиком means you can pick whichever translator you want from the group. If you said с каждым переводчиком, it would mean you have permission to speak to every single translator, one by one. Similarly, о любой проблеме means 'about whatever problem you choose', while о каждой проблеме means you will discuss every single problem on a given list.

Why use the perfective verb поговорить here instead of the imperfective говорить?

Поговорить implies a specific, completed action, often translated as 'to have a talk' or 'to have a chat'. Using the imperfective говорить here would imply an ongoing or general ability, sounding more like 'you generally converse with any translator' rather than 'you can go have a talk with a translator' about your current issue.