Which software tools from LatticeSemiconductor do I need to develop design with iCE40 LP FPGA?
If you ask Lattice, you'll need their ICEcube2 suite. The ICE 40 getting started documentation lists all that, I'll leave reading that bit of Lattice's website up to you!
However, if you ask the rest of the world: The ICEStudio is a free and open source IDE for these FPGAs. It uses the yosys synthesis suite, nextpnr or arachnepnr for place and route and supports a few programmers out of the box. It supports popular ICE40 devboards out of the box.