you don't need a true sine inverter for power tools, most of them are universal motors and will run just as well on DC. The amount of energy used by a power tool like a drill is very small, after all you don't run it more than a few seconds at a time typically. So a cheap 1000 watt "modified sine wave" (i.e., square wave) inverter will do the job, and your battery will be fine. The microwave is much more sensitive to the waveform, will run on the cheap one but not put out as much power by a noticeable amount. For the microwave you probably need at least a 1500W square wave inverter, which will do for occasional use. This will be a very heavy draw on the battery (say, 120 amps) and may be too much for it unless it is in very good condition and perhaps a gel or AGM (both of which have lower internal resistance than a flooded cell, with AGM being best). Or add a second battery in parallel. The energy usage of a microwave can be small (heating a cup of coffee - 1 minute @ 120A is only 2AH), or large (microwaving frozen dinners - 2 of 'em for 6 minutes each is 24AH).

If your main interest is power tools, I would suggest looking at battery powered ones. A nice 18V drill can do just about everything a small 110V drill can do, and you can charge it with a cheap $50 cigarette lighter type inverter if needed. I have a huge electrical system on my boat but I use a Milwaukee 18V drill and 18V sabersaw. Would buy their grinder too but I already had a 110V or two. All the tool companies also make a small circular saw and a sawsall, these work surprisingly well considering.