Valeyard is prosecutor in Gallifreyan, so since he said that name many times on "the trial of time lord", I'd say yes. He did speak Gallifreyan onscreen. Since the TARDIS translates everything, he may be speaking Gallifreyan all the time for all we know. Presumably, all the characters including him are speaking Gallifreyan in the episodes set on Gallifrey, but it is translated for the audiences benefit.
The TARDIS can not translate Gallifreyan. This means that the Doctor is in fact speaking English. However, it is likely that the two languages are near identical to each other as it has been seen other Time Lords speaking English with no confirmed/obvious translation ("The End of Time")
No, the TARDIS can translate Gallifreyan, it just takes it a moment to translate WRITTEN Gallifreyan, but spoken Gallifreyan is obviously being translated or other races who go there would have commented on the fact that they cannot understand what people are saying.
The above statement appears to be correct. What use would the Time Lords have for they universal translators inside their TARDISes, if the thing couldn't translate Gallifreyan for them.
Gallifreyan doesn't translate. There is no need for it to translate, it the technology used to translate languages is of Time Lord origin we can assume that everyone using this technology is Time Lord, meaning they can all speak/read Gallifreyan. Why translate a language you can already understand?