Use another magnet on the outside of the flask to hold the stir bar while you pitch the starter. You can buy coated magneic spinbar retrievers (looks like a swizzle stick) but it is just one more thing to stick in your brew, and they cost $10 to $20 from places like Cynmar
Regarding beechwood. The >>ONLY<< thing it does for the beer is to provide a neutral/sanitary substrate for yeast cells to sit on so there is more surface area to contact the beer as it conditions. They are pretty gnarly - rough, bent and twisted so beer circulates freely between them as they...
Will work fine. I placed a 40 watt light bulb on a ceramic base inside a #10 can to do the same thing in my garage fridge for fermenting and lagering during the winter. If you want to be really nice to your beer, put a 12v computer fan (powered by a wall wart) in the fridge to keep the air...