Learn Farsi: در حالی که دیگران تنها هستند، رابطه‌ی ما پایدار است. - While others are alone, our relationship is stable.

Not quite. While it does use ke to connect clauses, dar hâli ke is a fixed conjunction meaning "while" or "whereas" (literally "in the state that"). Unlike the duration expressions you are learning which express "it has been [time] since...", this just sets up a contrast between two present situations.

Could I use vaghti ke instead of dar hâli ke here?

You could, but it changes the nuance. vaghti ke is strictly about time ("at the time when others are alone"). dar hâli ke is used here to emphasize a contrast, functioning exactly like "whereas" or "while" in English.

Why does digarân (others) use the -ân plural suffix instead of the standard -hâ?

The word digar means "other". When used as a noun referring exclusively to people ("other people"), it takes the animate plural suffix -ân. You will often see this suffix used for groups of people, similar to dustân (friends).

Why is there an extra ی in râbete-ye mâ (our relationship)?

This is the ezâfe connector. Because the noun râbete (relationship) ends in a silent he (ه) which sounds like "e", we cannot just attach the usual short "e" vowel. Instead, we insert a "y" sound to link the words smoothly, which is written with the letter .