"We rarely run in the park."
Translation:Rzadko biegamy w parku.
Well, it's not wrong... but I don't know if it feels natural enough. It gives a strong emphasis on 'rarely', strangely strong...
Not really. It sounds to me like "The thing that we rarely do in the park is running".
The sentence you came up woth implies we do lots of other things in the park often, just not running. Sounds ok to me if it's part of a conversation where you explain what you and a buddy do in a park.
Biec is just as an alternative infinitive of biegnąć, which means to be running at the moment in a specified direction (both conjugate in the 1st person plural as biegniemy). This sentence however is about a recurring event, so the verb biegać (1st person plural: biegamy) should be used. Biegać either refers to habitual action (as in this case) or can also refer to a real time event, when people are just running around without a specified (common) direction.