Hola Aubreyrose! Hola Oopsy! Hola thrahtobe!
Sorry for not posting sooner but yesterday, I was working on ssilady's last-minute itin.
Aubreyrose, you will have a wonderful time in Barcelona! Many ladies (and some men) have traveled solo and written about their experiences. There's a small collection in the Top Q - How do I travel alone?
As far as eating alone, it's not a problem really. And you probably won't be alone for long. Folks are friendly. On our last visit to Bar Pinotxo, I met a visitor from Australia. He didn't know what to order so I gave him some of our food and made some suggestions as well. I haven't been to Cal Pep but a friend who eats there on every visit has also mentioned that lots of food sharing amongst strangers is typical.
yesiloves2travel is a foodie who visited and ate her way through Barcelona. She was watching her budget too. You can search this forum for her trip report.
It IS a bit challenging to combine sightseeing and eating but since you use Google maps and don't mind researching, you'll manage very well. If you haven't already added the tapas bars to your own Google map, you could use the Tapas Map linked in the Top Question. It contains almost all of the places you listed above PLUS the most popular sightseeing landmarks. All you would need to do is check the hours of operation for the sight and the bite. :-)
We each have personal favourites. I like the Palau de la Musica Catalana as well as Palau Guell. It's the 1st residential bldg in Barcelona designed by Gaudi and it was done for the family of Count Eusebi Guell. He's more famous for Parc Guell. Palau Guell is off of Las Ramblas, near the waterfront. Closed on Mondays.
With the exception of Palau Guell, the Gaudi landmarks in Barcelona are open daily. The Picasso Museum is closed on Monday. The Sagrada Familia is busy all the time, especially in the mid-morning (definitely buy tickets in advance or go in the late afternoon).
Re: Runnerbean Tours, I would take the Barri Gotic tour. The guides know lots of interesting facts about various bldgs and they'll make sure you see the important sights. You can email them to let them know you're coming or just show up, on the day of. There is a maximum number of people they can take at one time.
Re: Montserrat, imho, the best time is when the weather is good for hiking in the mountains and the Escolania is singing (they only sing for 10 min or less but it's still a beautiful thing to experience). You can check the Escolania schedule here:
Re: Montjuic and Magic Fountain. The fountain is operational at night and only certain days of the week. The schedule is in the Top Q. Recently another TA member visited and commented that it was more beautiful than similar shows in Las Vegas! I wouldn't combine it with daytime Montjuic activities since it's really closer to Placa Espanya and it's at night. You could easily get there on the metro from the centre of town. My personal favourite combination is snacks at Lolita and Quimet i Quimet, followed by the Magic Fountain. Go to Q i Q first (preferably at opening, 7 pm). It's really small and can get very crowded, very fast.
Re: the cable cars. The Funicular de Montjuic is not scenic since it goes up INSIDE the mountain. The Transbordador Aeri del Port is very scenic. I like going from Montjuic to Barceloneta. If you like modern art, you could try this itin:
1) Funicular de Montjuic UP
2) Walk over to Joan Miro museum
3) Walk 10-15 min along Av Miramar to Transbordador Aeri del Port
4) Soar across the city to Barceloneta
5) Snacks at La Cova Fumada, Can Paixano
6) Check out nude sunbathers or nude cyclists (ouch)
7) Walk back through El Born to Santa Maria del Mar
8) Euskal Etxea and Museu Picasso
A FEW MORE COMMENTS ABOUT YOUR FOOD LIST:
Bar Mut is a bit pricey and as a solo diner, you may have some trouble ordering too much. I've never heard of Reserva Iberica. If you are looking for jamon iberico, you can get little tastes from the vendors at the front of the Boqueria. The really good stuff is sold in the Club Gourmet of El Corte Ingles department store. 16 euros for a pack of Joselito or Cinco Jotas (5J).
I love Formatgeria La Seu too! Pls say hi to Catherine for me and tell her I'll be there soon! Her hours are very traditional since she's usually working alone. Closed from 2-5 pm and Sundays, I believe. Don't miss her cheese ice creams and her other business, Bar Zim. Francesc makes the best pa amb tomaquet, serves delicious Catalan cold cuts and offers a nice selection of local wine by the glass.
Feel free to work on your itin here. Oopsy and I will be glad to help.