Are samurai still trained in Japan? Are there samurai schools?

The short answer is, No.
Samurai were a caste or class in a social system in Japan. The system started its end in Japan in 1868 with the formation of the Meiji government. In 1871 the domain (han) system was abolished, and samurai were forbidden from wearing the famous top knot, and a few years later, in 1873, payments made to samurai retainers were stopped. From that point, samurai had no stipend to survive on as before. That same year the new government created a conscription service for the national military. In 1876 the Meiji Government issued a ban on carrying swords. The Meiji restoration in 1868 saw the start of the modernizing of Japan, and the end of the Tokugawa Shogun rule.

It is important to note that non-samurai were heavily involved in martial arts of the 1800s in Japan. Schools before 1868 are often referred to as koryū 古流 or kobujutsu 古武術 (classical schools).
These schools still exist. The older systems of samurai education, however, do not. The large schools where samurai were educated in their domains and taught many different arts do not exist anymore. There are no holistic samurai training anymore. Many various arts are taught in different schools, however. sword katana iaijutsu