cbixler89, most of your questions can be answered by the Titlis website: http://www.titlis.ch/en
- Is it open - AFAIK yes, I can find no reference on the website to a closed season.
- Will there still be snow tubing? - Much the same answer. See here: http://www.titlis.ch/en/glacier/glacier-park "All kinds of fun snow activities await you all year round at our glacier park."
- Expense - Unfortunately Switzerland is expensive. In this case you get what you pay for. It is world class. I do not know what "AAA, USAA, AARP" are, but I would not expect anywhere in Switzerland (or any other country) to give discounts for foreign organisations. The one thing you do get reduction for is Swiss railways ½price card and other passes. See here for prices and reductions: http://www.titlis.ch/en/tickets/cable-car-ride
- Better than Titlis? Possibly the Jungfrau, that has a much bigger glacier to look down on, but no additional activities. Also possible and cheaper are the nearby Pilatus and Rigi. Both much lower peaks.
"I plan to rent a car for the day, but I am open to taking the train" - From Basel I would recommend the train. This is easy, one hour Basel To Luzern, and a further hour to Engelberg, which is the bottom of Titlis where the road ends. Any few minutes you save with a the car would be more than lost in the time taken to pick up and return the car, refuel it, finding your way, parking etc. And the view from the train between Luzern and Engelberg is good.
Look up times and prices from Basel to Titlis (via Engelberg) on the Swiss railways site: http://www.sbb.ch/en/home.html
"free time the second day before the ship leaves" - exactly how long? When do you have to be at the ship. You might have to time this closely to do it in one day and get to the ship in time.
WEATHER - do not do this if the top of the mountain is in cloud, you will be disappointed. But you could go to Engelberg and part of the way up. When you get up in the morning, check the weather (ask the hotel) before making a final decision.