This is a question on an active area of research, with lots of work on it (for the general case of algebraically closed fields of char zero). It is historically and conceptually closely connected to Kaplansky's conjectures, especially the 6th one. There are detailed results on the classification for $dim\leq 60$. A detailed list of references might be very long. I list below some classic surveys up to 2014. (Sorry in advance if you are already aware of these).
- On the semisolvability of semisimple Hopf algebras of low dimension, Sonia Natale, 2006
- Classification of Semisimple Hopf Algebras, Handbook of Algebra
Volume 5, 2008, Pages 429-455 Akira Masuoka
- ON THE CLASSIFICATION OF SEMISIMPLE HOPF ALGEBRAS: STRUCTURE AND APPLICATIONS, 2010, Sebastian Burciu
- Classifying Hopf algebras of a given dimension, 2014, Margaret Beattie, Gaston Andres Garcia
- ON KAPLANSKY’S SIXTH CONJECTURE, 2014, Li Dai, Jingcheng Dong
The leading idea (in my understanding of course) is to classify semisimple Hopf algebras in terms of group hopf algebras, their duals and twists of them. This has been implemented through two distinct "strategies": One is based on using the theory of Hopf algebra extensions and the other (which is more recent) on the classification of the fusion categories associated to semisimple hopf algebras. These are nicely reviewed in ref 3.
I am out of office right now, so I have very limited access to resources and cannot provide much info on more recent developments but here are some (hopefully helpful) comments regarding the references provided and your partial questions:
- Regarding your Q1: In reference 1, the main result is the following theorem:
Let H be a semisimple Hopf algebra of $dim< 60$. Then H is either upper or lower semisolvable up to a cocycle twist.
A survey of the literature is also included together with an important amount of results obtained by the author. In p. 5, there is a table gathering results on the classification of $dim\leq 60$ based on the factorizations of the dimension in terms of prime numbers. A complete classification is included for dimensions $30$ and $42$. In refs 2 and 3 further results (obtained after 2008 are presented).
Ref 4 is also worth to be mentioned separately: it is a comprehensive work which although does not focus on the semisimple case -rather the opposite- it contains newer developments and a very interesting table at the end (see p. 23) which in some sense may be seen as complementary to the table of ref 1: In there, the authors classify open cases for $dim\leq 100$, distinguishing between the semisimple, the pointed and the Chevalley hopf algebras.
Regarding your Q2: If we are speaking about a hopf algebra in general (not necessarily semisimple), over an algebraically closed field, this is essentially Kaplansky's 6th conjecture. This is a long story and there is a lot of literature investigating it.
Even if we confine ourselves to the semisimple case, the question (to the best of my knowledge) is open in general. However, there are several results. Here are some things I can recall:
- By an old result of Frobenius, the answer is "yes" for semisimple group hopf algebras (hence when the char of the field is zero) but it is "no" for the general case of group hopf algebras. (A proof for semisimple $kG$ and counterexamples for non-semisimple $kG$ can be found at: "Methods of representation theory-with applications to finite groups and orders" by Curtis and Reiner). For this reason, semisimple HAs which satisfy this property are frequently called to be of "Frobenius type" in the literature.
The answer to your Q2 is also affirmative -so we have HAs of Frobenius type- in the following cases (I am confining to the case where $H$ is assumed semisimple; there are also works where semicimplicity of $H$ is not an assumption):
- When the dim of the simple module is $2$ (Nichols and Richmond have shown in 1999 that semisimple HAs, over an algebraically closed field, with a 2-dim simple module must be even dimensional).
- When the semisimple HA $H$ is of the form $D(B)$, with $B$ fd and semisimple. This has been shown by Etingof and Gelaki in 1998.
- When $H$ (apart from semisimple) is also cosemisimple, quasitriangular and over an alg closed field. This is again a result of Etingof and Gelaki in 1999.
- Ref 5 focuses on the investigation of Kaplansky's 6th conjecture for the semisimple case and provides some more recent developments.
Regarding your Q3: in dimension 36 the only simple example is a twisting of a finite group. This is obtained in 1 together with some interesting results on values of dimensions where semisimplicity excludes simplicity (in the sense of hopf algebras of course). This is the only thing I can remember right now but I believe there have been newer developments since.
Hope that these are of some interest to your purposes.