Which software a dubbing studio use for subtitles and closed captioning workflow?
Most episodic (network and cable) TV is captioned by service bureaus who use proprietary software. Some of the big players are VITAC, NCI, and WGBH, and there are several more. They all provide various levels of service, and some rely more heavily on automation / voice recognition etc than others. This is all offline captioning and the goal is accuracy in copy, placement and timing.

The other main thread (and skill) is live writing or real-time captioning, which is done with stenowriters (chord typing) as used by court reporters. These are always delayed by some time because of the need to hear, interpret and input. It's almost always done remotely, with the caption writer networked into a local encoder. The same bureaus usually offer both live and offline.

To learn more, check out the links above -- and google for more -- then maybe contact the providers that look interesting to you.