Takeoff minimums are going to be directed by each company's individual ops specs. Standard takeoff minimums are 1sm visibility (5000 RVR) for two engines or less, and 1/2sm vis (2400 RVR) for three or more engines. Below that, each individual carrier must apply for and be approved for lower departure conditions.

As a general rule, the lowest that most carriers can depart is usually 600/600/600 (RVR), provided that you have the required visual and lighting cues (runway centerline lights and markings). As stated before, this will vary depending on each company and the authorizations that they have. FAA ops specs C056 and C078 cover the applicable requirements. This section of each company's ops specs will spell out the various conditions (weather and available visual cues and equipment) necessary for lower minimums.

A takeoff alternate is required any time that the weather at the departing airport is below the minimums for a suitable approach that can be used to return to the airport in the event of an emergency (based on current conditions).