Learn Finnish: Onko varmuuskopio yhä nähtävissä palvelimella? - Is the backup still visible on the server?

This is the passive present participle of nähdä (to see) in the plural inessive case. Literally, it means something is in the state of being able to be seen. You will see the -tavissa or -tävissä endings used frequently to express availability or detectability, translating naturally to English words ending in -ible or -able, like visible, accessible, or available.

Could I use näkyvissä instead of nähtävissä in this sentence?

You could, but there is a slight difference in nuance. The word näkyvissä simply means something is visually in sight, coming from the verb näkyä (to be visible). However, nähtävissä comes from nähdä (to see) and emphasizes that something is available to be looked at or inspected by someone. For a digital file like a varmuuskopio (backup), nähtävissä sounds more natural because it implies a person can access and examine the data.

Why does the word for server, palvelin, become palvelimella?

Nouns ending in -in are often instruments or technical tools, like puhelin or avain. When you add a case ending to these words, the -in changes to the stem -ime-. Therefore, palvelin becomes the stem palvelime-, to which we add the adessive ending -lla to get palvelimella (on the server).