Ok, I am a former hotel employee of several years (too many if you ask me!) and all hotels I've every worked at or been in contact with have the same policy regarding credit cards.
You MUST reserve your room with a credit or debit card. You simply must, it's just not an option. Depending on how you book your hotel, ie. over the phone directly, through Priceline,
AAA, etc. the card may be charged at the time of booking.
If you use your Mom's credit card for airline tickets, then I suggest you kindly ask her if you can use it again to reserve the hotel. Make sure that you are clear on what the policy is at time of booking. If you call the hotel directly, they will only ask for the card# to reserve the room for you.
Upon arrival at ANY hotel, the card you give them at the desk is the one they will "pre-authorize" for incidentials. You can always give them a different card at check in then the one you used to reserve. Depending on the hotel and the level of services this amount varies. It's typically $10 per day at places like Hojo's, Holiday Inn, Best Western etc. and $100 per day at your more "fancy" hotels. So if you're staying at the Holiday Inn for 5 nights, they would typically pre-authorize $50 for your stay.
Sometimes hotels have a very rigid "no cash" policy and will not accept a cash deposit instead of a credit card for incidentials. The idea behind this is that if you go crazy and trash the hotel room, then are able to charge it to your credit card. If all you've given them is $50 cash, then they are out of luck! Check with your hotel directly regarding the cash policy and incidentials.
All hotels will allow you to pay in cash upon check out. However, most will not accept your cash payment upon check in and you will have to wait until you leave to settle your bill.
So as I suggested earlier, ask your Mom to let you reserve the hotel room with her card. Nothing will be charged if you book directly with the hotel or online at their website but be careful for places like Priceline, Travelocity where you might be charged upon booking.
Once you arrive at the hotel, if you haven't already asked them about their cash policy, see if you can put down a cash deposit for your incidentials. If not, just use Mom's card and just settle up when you leave. Nothing is charged to the card until you check out. A set amount is pre-authorized, meaning the hotel is "holding" that amount but once you've checked out, that amount is either charged towards your final bill or released if paying cash. BTW, if you do pay cash at check out and have used Mom's card for the incidential pre-authorization, nothing will show up on the credit card bill. You/Mom will never see this pre-authorization amount. Same if you pay with the card upon check out, you will only see the final bill amount.
I know this doesn't answer your real question about hotels that you don't need a credit card. To be honest, there aren't many hotels out there that don't require a credit/debit card at least to reserve. And frankly, would you want to stay at one that didn't require this??

The policy is a good one, so I suggest you try and work around it if you can.