With an engine shutdown, if the APU is already on, I'd close the engine bleeds and open the crossbleed. The checklist (precaut. engine shutdown) calls for starting the APU if available, so once the APU is on, I'd close the bleeds.

Personally, If performance doesn't require bleeds to be closed (cold or low field elevations), or if the wx is nice, I shut down the APU. If the WX is bad, I leave it on just in case and shut it down around 10,000 feet. If the APU is preferred to be shutdown prior to takeoff, you can still do a bleeds closed takeoff, you just turn off the packs and close the bleeds until 400 feet or whatever your acceleration height is for that runway. You'll just be unpressurized for that period of time. Not a big deal. Closed bleeds saves engines, allows them to run cooler, more power, and reduces wear and tear at high power settings.

Shutting down the APU lengthens it's life, and saves 130 pounds per hour of fuel. The APU burns a slight amount of oil, and it only holds 2 quarts. Running it constantly under load (bleed and gen) eventually leads to low oil quantity. Treat it well, because once you're without it, life becomes much more difficult and environmental conditions become much more difficult to deal with!