The train is a lot faster than dealing with airports. I cannot explain why you're not seeing availability. Have you tried the day before your travel date and the day after? If those are available, I'd be concerned that there's something really big happening in one of the two cities and a bunch of trains are sold out. That seems very unlikely this early, however.
Note that Parc Guell is outdoors and not totally level. There's a fair amount of walking. Unless the weather is really mild, it's best not to go there at midday.
I'm not a fan of the Ramblas. It's very crowded and has lost much of its charm. I think that's walking your mother can do without. If you do go, be very wary of pickpockets; it's apparently one of their favorite places.
Another vote for Sant Pau. Not only is the multi-building complex lovely, it's a place in Barcelona where you can just walk up, buy your ticket, and go inside. It's very difficult to plan short trips to Barcelona since you have to pre-purchase tickets to so many places to avoid long lines, which requires you to guess how much time you'll want at each place. These are my rough time suggestions. They do not include travel time from site-to-site, and they do not include time you may spend in line even with pre-purchased tickets.
La Sagrada Familia: 90 minutes without a tower
Parc Guell: depends on how much territory you want to cover, but 90 minutes might be reasonable.
Casa Batllo: 90 minutes (expect massive crowds; might be better in the evening)
Palau de la Musica Catalana: mandatory tour is a fixed length, probably about 1 hour
Sant Pau: 2 hours (it's large, and I would have spent more time there if they hadn't been closing. No need to buy in advance.
