Yes, a nonprofit organization may create a subsidiary with either a for-profit or a nonprofit structure. In some situations creating a subsidiary may make sense. If you think this is something your organization should do, please talk to an attorney familiar with both corporate and nonprofit law to fully understand the tax and legal implications.
Technically, yes. However, your initial support probably will not come from foundation grants since most institutional funders generally require proof of 501(c)(3) status and prefer to support organizations with a proven track record of fiscal responsibility and programming successes.