Are these points about Easter true?

Answer:

A monk named Bede, wrote around 700 A.D.:

"Eostur-monath has a name which is now translated Paschal month, and which was once called after a goddess of theirs named Eostre, in whose honour feasts were celebrated in that month. Now they designate that Paschal season by her name, calling the joys of the new rite by the time-honoured name of the old observance."

The existence of such a goddess is a bit obscure. "Eosturmonath may simply mean 'the month of opening', appropriate for a time of opening buds" [Adrian Bott, "The modern myth of the Easter bunny," The Guardian, 23 April 2011]. So whether the name came from the name of a spring month or the name of a now unknown spring goddess is debated. What remains is that name became attached to Christians' celebration of Jesus' resurrection in the spring.

Of more interest is why Christians decided to start an annual celebration of Jesus' resurrection. What history tells us is that in the second century, Christians started wanting festivals similar to what the Jews had. "The New Testament contains no traces of annual festivals; but so early as the second century we meet with the general observance of Easter and Pentecost, founded on the Jewish Passover and feasts of harvest" [Philip Schaff, History of the Christian Church, Third Period, AD 311-590, The Church Year]. Only early on it was called "Pascha" in Greek, which is the Greek word for "Passover." Initially, it was held at the same time as the Jewish Passover, but in A.D. 325, the Council of Nicaea set the date to the first Sunday after the first full moon after the Spring Equinox. In other words, the entire holiday was started because people were not content to simply do as God directed.

Easter eggs were a borrowed tradition from medieval traditions in Europe. "Decorated eggs had been part of the Easter festival at least since medieval times, given the obvious symbolism of new life. A vast amount of folklore surrounds Easter eggs, and in a number of Eastern European countries, the process of decorating them is extremely elaborate" [Katie Morrow, "Why Easter is called Easter, and other little-known facts about the holiday," 29 March 2021]. "Decorating eggs for Easter is a tradition that dates back to at least the 13th century, according to some sources. One explanation for this custom is that eggs were formerly a forbidden food during the Lenten season, so people would paint and decorate them to mark the end of the period of penance and fasting, then eat them on Easter as a celebration" ["Easter Symbols and Traditions," History.com, 24 March 2021]. Where the medieval Europeans picked up their practice is more debated, but decorating eggs has been around for a long time.

The Easter bunny dates back to a 17th-century German tradition. "According to some sources, the Easter bunny first arrived in America in the 1700s with German immigrants who settled in Pennsylvania and transported their tradition of an egg-laying hare called “Osterhase” or “Oschter Haws.” Their children made nests in which this creature could lay its colored eggs. Eventually, the custom spread across the U.S." ["Easter Symbols and Traditions," History.com, 24 March 2021]. Again, where the German's got their tradition is debated, but it is not hard to see why rabbits are associated with the spring season.

Regardless of the origins, the traditions may be fun for children, but they have nothing to do with New Testament Christiainity.