I'd be very surprised.
No it can't come alive.
That snowman would be Frosty.
That movie is the original "Frosty the Snowman" movie as aired in 1969.
During a heat wave, a snowman would melt due to the increase in temperature. The snow that makes up the snowman would turn into water as it is exposed to higher temperatures.
No
It was Pigpen: when he played in the snow it became grey, thus the greysnowman.
how might the particles that make up a snowman end up in the ocean
If I build a snowman in someone else's garden who owns the snowman is a good question. Probably you do, since snow is a natural phenomenon that falls everywhere and it would be difficult for the other person to claim it as their property by law just because it happened to be in their garden. However, the craftsmanship and artistry that you'd put into building the snowman would mean that it was your own creative work, and so it would probably belong to you.
Because he would melt in the spring.
It is from a song made in the 1950's and became very popular. From the song came the story.
It depends upon the desert and season of the year. A snowman could survive almost indefinitely in Antarctica. A snowman would only survive a few short hours in the Mojave or Sahara during the summer.
You could call a snowman who refuses to melt a "resilient snowman" or a "stubborn snowman."