Does this solar panel setup with batteries meet my power needs?
You're confusing yourself with amps and volts, so it would probably be a good idea to just stick with watts, and convert at the end.

You say your total load is 35 W, so you need 35 W &times; 24 h = 840 W-h per day of operation. I would recommend having enough battery capacity for at least 3 days of operation, or about 2500 W-h. (This would be 100 A-h at 24 V.)

You'll want the batteries to recharge within a day, even if they're substantially depleted. Battery efficiency is on the order of 70%, so you'll need 2500 / .7 = 3600 W-h of energy for the recharge. If you get a minimum of 6 h of sunlight in a day, you'll need 3600 W-h / 6 h = 600 W of panels.

So I would say that your estimates are very much on the low side. You can either expand the power system, or find lower-power devices to run the application.