Hillel and Shammai represent the last of the Zugos the Pairs of Tannaim. After them, this period ceased and the next generation could be considered a new period. You could say a transition period between two individuals and having schools without representing one or other person. So in this respect then there was no such thing as Beit Rebbe Akiva because he didn't represent one school, but a combination of what he learned from his teachers. There was no longer a single path through a single teacher representing a single tradition. Rebbe Akiva represented the sum-total of the Oral Torah with different teachings melded into one. If you follow Rebbe Akiva's line of teachers, he learned from both the academies of Hillel and Shammai.
After Hillel and Shammai passed on, their students continued their teachings. Their students represented their academies and the disagreements between them continued and increased. The Oral tradition, however, as a complete unit could no longer be encapsulated by even a single academy and that could not be passed on, from one generation to the next, but rather required students from both academies to learn and transmit the tradition.
So to answer the question Beit Shammai is the academy or school where Shammai's teachings and Oral Torah was passed down and interpreted. The same is true for Hillel's Academy. Subsequently, the teachings could no longer be contained in a single academy representing the Rebbe's transmission. You had to go to different Academies to receive and transmit the Oral Torah to your disciples. Thus the age of a single 'school of thought' was over. After that, no teacher or disciple could claim that the sum-total of the Oral Torah he received was from a single particular school, but rather from a conglomeration of different teachers/schools.
As the OP mentioned, the Braisos - which are Tannaic sources mention Tanna d'Bei Rebbi Yishmael. But it is not common. The Gemara is a combination of Aramaic discussion and Tannaic sources.
To answer the second question, when it is mentioned in the name of someone else, that is the originator of the Oral statement. However, when Beis Hillel or Beis Shammai is the originator, it cannot be directly linked to Hillel or Shammai themselves, but rather the academy in which they taught.