Can I use a solar panel, a lawn garden battery and some sort of copper wire to warm my garden soil for earlier planting? Answered

No. You need far more power than you can collect from a solar panel, and a copper wire isn't what you need to generate heat. Rig some "hoop greenhouse" things or learn about cold frames and other cold-weather gardening methods.

You are better off using the thermal properties of sunshine, especially if cost is your main concern. Black cloth will dump more energy into the dirt than a solar powered electric heater will, at least at the level you are dealing with.

try to check out the BBC show "It's Not Easy Being Green" Season 1 - Episode 2...a homemade solar greenhouse heat sink is made and is really ingenious & fairly inexpensive...or do a google search on "solar greenhouse heat sink", there are numerous sites...here's 1; solar green househouse heat sink