To answer your second question, yes. The Japanese Emperor Go-Daigo attempted to regain power and make the Japanese monarchy a Chinese-style dictatorship, during the Kenmu Restoration of 1333-1336. There were also attempts by two Retired Emperors (that is, Emperors who had abdicated) to gain power for the imperial family, though not the Emperors themselves. Go-Shirakawa worked with the Minamoto clan in the Genpei War of 1180 to 1185. His grandson, Go-Toba, attempted to gain power in 1221. Both had abdicated decades before their attempts.

For your third question, no Shogun, Daimyo, or outside family ever succeeded in deposing the imperial line, at least not since Japan started writing reliable historical texts in the 8th century.

For your fourth question, imperial authority and influence had waxed and waned in the millennia before Meiji. But for the preceding 500 years, the Emperors of Japan had little practical power.

Dating back to the Heian the Fujiwara set up a system wherein an emperor would be appointed at 10 or so, marry a Fujiwara or two, sire an heir, and then abdicate and become a monk in his twenties. (At which point the cycle would repeat.) Because the emperor would always be a child under the thumb of his Fujiwara grandfather and father(s)-in-law, and would retire before he had any real authority, the emperor himself never had any power and political struggles focused more on who would hold various ministries than on who would be Emperor. This meant that it would be both superfluous for the shogun to depose the emperor, and also absurd (since the shogun could not claim descent from Amaterasu in the male line).

The Fujiwara lost their power and eventually political power passed from the court aristocrats to the military class, but the same basic pattern of fighting for the most important ministry was in place for a millenium.