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tanhâ is flexible and can act as an adjective meaning 'alone' or an adverb meaning 'only/solely'. Its placement right before the compound verb esterâhat mikonim strongly emphasizes the action, making it mean 'we solely rest' (as opposed to working or studying). To unambiguously mean 'we rest alone', you would typically use a phrase like be tanhâyi.

Does the placement of tanhâ change exactly what is being restricted in this sentence?

Yes. Because tanhâ is placed right before the verb phrase esterâhat mikonim, it restricts the action: resting is the exclusively permitted activity. If you moved it to the beginning of the sentence (tanhâ mâ...), it would restrict the subject instead, meaning 'Only we rest in this room' (and nobody else does).

The meaning is restrictive, much like 'we do nothing but rest'. Why is the verb affirmative?

When using tanhâ or serfan to mean 'solely' or 'exclusively', Farsi always requires an affirmative verb. You only use a negative verb for restriction when building an exception structure with words like joz or magar (which translates closer to 'we do not do anything except rest').

Is it possible to place tanhâ inside the compound verb, saying esterâhat tanhâ mikonim?

No, you should avoid splitting a compound verb like esterâhat kardan with an adverb of restriction. tanhâ must come before the entire compound structure so it can modify the whole action smoothly.