I've read that solderless breadboards cannot handle more than 1 A of current or so. What kinds of alternatives exist to a solderless breadboard? Would a soldered board make a difference if I want to handle something like 5 A of current? What other options are there?
If you just need to work fast, I recommend a peice of 1/4" plywood as a project surface. Get a variety of lengths and sizes of wood screw and screw a breadboard, some screw terminal strips, heat sinks, electrical boxes, power supplies, etc (hereinafter Resources) to it. Don't overthink the layout, just make sure it's big enough and has enough Resources. You can fit a lot of stuff into 18"x18". What you're trying to do is create a work area where you're not plagued by bad connections, congestion, and missing Resources.
You should be concerned with fire when working with high currents. Just make sure you can easily disconnect power and keep a working fire extinguisher nearby.
Plywood is flammable. You could use metal but metal is worse in almost every way for this purpose other than flammability. There's usually lots of stuff that can catch fire on a workbench, so you need to heat sink the hot chips properly anyway.
If the high current path is a relatively short or simple one, you can connect the leads of the components in the high current path with a heavier gauge air wire, shorting the path through the breadboard. This lets you keep the breadboard in place as an anchor for those components and still gives you the option of low current circuit pathing for those leads which carry lower current.
Just make sure that the entire high-current path is covered with the heavier wire. This includes the supply and ground leads -- don't make the mistake of relying on the +/- power strips on the side of the board.
This technique is obviously most helpful when dealing with e.g. DIP, TO-220 package transistors which have a 2.54mm pitch.