Here are Park tools for the BB. The HCW4 has pin spanners on one end and the fixed cup hole on the other. The HCW5 has a headset wrench (32mm) on one end and the BB lock ring spanner on the other. Using a vise for the fixed cup is good, and other pin spanners look like tongs or the end of a coat hanger with bent tips. You'll have to decide if you can scrounge the tools or let the LBS do it.
If the BB is in good condition, there is no problem with reusing it, but they are not sealed very well and can be tricky to adjust. One sealed replacement with a square axle is the Shimano UN-54, about $25. Beware the plastic cup, don't overtorque or it cracks. Test thread first, as you may need to chase or deepen the threads. This BB needs the Park BBT-2.
If you bang on the stem bolt, yes, put the wheel back in and the fork back in the headset/lightly reassemble. Make sure the stem bolt is at least partially threaded into the wedge before putting a block of wood on it and whacking it with a hammer. Occasionally I have seen folks put handlebars back in, and straddling the wheel while twisting a bit to loosen the stem.