Jedi Youngling - This is the unofficial title given to the Jedi Initiates, young children taken into the Jedi fold and taught the fundamental basics of Jedi teachings. These clans are taught by different venerable Jedi Masters. Yoda had lead over the Bear Clan. They are not considered Padawans at this time.
Jedi Padawan - A Jedi Padawan is a Youngling who has come of age, or shown exceptional knowledge and skill , and is now paired with a Knight or Master to complete their training one-on-one. A Master or Knight can have only one Padawan at a time. Anakin Skywalker is a rare example of one who never studied in a Clan.
Jedi Knight - A Jedi Knight is a Padawan who has completed their one-on-one training with their mentor, and successfully passed their Jedi Trials. Upon passing their trials they are Knighted and may now take on their own Padawan for training, but this is not a requirement.
Jedi Master - This title can be achieved in different ways. The most common manner inwhich a Knight may become a Master is to have successfully trained their first Padawan to Knighthood. Upon the Padawan becoming a Knight, the mentor is then dubbed a Jedi Master. Another method is to attain the title by honorary means. This means that a Knight who has not trained a Padawan may still show a deep knowledge and understanding of the Force, or do a great deed, to the extent that the Council may find them worthy of the title. Examples of this route include Masters Atris, Saesee Tiin, Fay, Stass Allie and Depa Billaba (no padawans are ever mentioned for the last two).
Jedi Council - For the most part, it is from the pool of Jedi Masters that the Jedi Council members are chosen. The High Council is composed of 5 lifetime members, 4 long-term members, and 3 short-term members. It should be noted that to be a Master does not mean one is, or required to be, an Council member. Or vice versa. Though Masters are normally chosen to sit on this body, it is not uncommon for a Knight to have a place. Notable Knights to have had seats on the High Coucil are Ki-Adi-Mundi, Stass Allie, and Anakin Skywalker (the first two evetually attained the rank of master).
** It is also of worthy note to comment that desperate times call for desperate measures. There are instances in the history of the Jedi Order where rules and regulations have been overlooked for many different circumstances. But these rare measures are not to be taken as the norm, as they are sometimes done without even the entire council's approval. **
On topic now; yes, Dooku was Yoda's Padawan in the "one-on-one" sense. Just as Depa was Mace's Padawan, Qui-Gon was Dooku's and Anakin was Obi-Wan's.
But yes, it is possible that Dooku may have been in the Bear Clan and trained under Yoda in that sense as well.