In my experience, local businesses are often solicited to sponsor/donate to the running of competitions by the LOC. In some cases, the club may offer program ad space at no cost in exchange for a donation toward the running of the competition or provide other incentives. Many businesses also donate towards the competitor gift bags or that sort of thing.
Getting a major sponsor (like, a title sponsor) for an event such as ANs would almost certainly have to go through U.S. Figure Skating. It could not be a sponsor that conflicts/competes with any of the major USFS sponsors if it was not one of the major sponsors. Maybe a local business could make a large donation and get some recognition, but that would have to be approved way in advance, and would obviously only apply to the event at that location - the next location may not get the same local support.
The reality is, major sponsors are going to put their name on events that have exposure to a wide cross-section of the general public, and that have the potential to make them more money (for example, by sponsoring an up-and-coming athlete that has the potential to go to the Olympics, which will bring their brand more visibility). With adult skating, that longevity and exposure for a corporate sponsor isn't there. No tv coverage, no ads, no real exposure outside of the adult and/or skating community. That actually means sometimes even local businesses don't want to sponsor or donate to the event, because they don't see the long-term benefit.
Regarding the entry fees, that is actually a highly discussed topic right now since registration is open.
Per the 2016 SWP Regional announcement, entry fees for the kids were $175, plus a $10 admin fee. For singles skaters who qualified for Sectionals, an additional $175 entry fee and $10 admin fee was required. Pairs and dance started at Sectionals, so they did not have the second $175 fee.
This year, at Adult Sectionals, qualifying is $175 plus the $10 fee. If you qualify, you do not pay another entry fee for the qualifying event at ANs. Non-qual at Sectionals is up to the LOC: For example, for Pac Coast it's $135 first event, $100 for each subsequent (plus admin fee for entryeeze entry). For Mids, it's $130 for Non-qual IJS first event, $70 for each subsequent non-qual IJS, and then $100 for first non-qual 6.0 event, $60 after that.