Barcelona - Too Much To Do and Too Little Time!

Well, you could do it all, but it would take months, or a lifetime.
Your 5 nights means 4 full days, and with basically a whole or most of a day for Montserrat, you have 3 full days left for Barcelona itself, and your plan for those 3 days seems reasonable, assuming you'll start with your most important sight and work down your lists.
Have fun!

The timing of the food tour is unfortunate, because it probably blocks you from buying a first-time-slot ticket to one of the sights where advance-purchase is necessary to avoid lines. You could, however, do a Barri Gotic walk, go to the cathedral, or maybe see Santa Maria del Mar (I haven't checked its hours).

I don't think you need to pre-purchase a ticket for the Sant Pau Modernista site.

Casa Amatller has a modernista façade but a sort of pseudo-castle interior. It's not uninteresting, but it might not be what you're expecting. (Then again, from your list it appears you have done a lot of research!) I've not seen comments suggesting that you need to buy tickets in advance, but maybe you do. It's right there in the same block as Casa Batllo, so I can well imagine people getting turned away at Casa Batllo and deciding to see Casa Amatller instead.

Some of the modernista sights are open into the evening, which will help stretch your days if you have enough energy to take advantage of those extra hours.

As for your arrival night, I recommend not buying an advance ticket to anything, because you may not be in any condition to enjoy it. The usual recommendation is to stay outside in the daytime (perhaps after a short nap) to begin vanquishing jetlag. I have never, over the course of perhaps 15 trips to Europe, been awake enough to do anything on my first evening, and I wouldn't risk any money in an effort to do so. Entry fees in Barcelona are generally very high.

I don't know whether you're aware that they eat both lunch and dinner very late in Catalunya. I'm not sure the locals would consider the time before dinner "evening", and for them dinner starts at 10 PM. So I'm not sure what time of day you might take your breaks (not that it necessarily matters).

La Sagrada Familia - the advice I was given, followed, and pass on: get tickets for the earliest entrance, line up 15-20 minutes in advance and be among the first to enter. The church is relatively empty and quiet, a good time for nearly people-less photos and quiet contemplation and appreciation of the nave. I thought the audio guide was worthwhile and what I enjoyed most about seeing one of the towers (I still don't know which is which) is the walk down a beautiful staircase. After the tower, around 10.30 the sun was at the right angle to light the interior through the stained glass and the colors on the floors and walls were striking. So I don't think you have to wait for sunset. BTW sunset in early May is around 9.00 pm so it's not feasible anyway. From there it's an easy 10-15 minute walk to Sant Pau with lots of cafes along the way if you want a drink or lunch. I wandered independently and loved it. Allow enough time, since it's a large site.

Take one of the walking tours of the Barri Gotic. The TI's tours are generally very good. Runnerbean also has a very good reputation. I took a rooftop tour of Santa Maria del Mar that was excellent.

It's a long way from the top of Montjuic to the Magic Fountain at its base. I wouldn't want to walk it after dark. Most of the views from the top are obscured by the darn trees. I think the really good views (which I didn't see) are from the fortress walls, access is from inside the sight. The easiest way to the fountain is from Plaza Espana - a short walk, then up the stairs or escalators.

Thank you all so much.

acraven, yes, we would have preferred an evening food tour, but all the others focused mainly on tapas and wine. We don't drink, so those wouldn't have been ideal.
Thank you for the reminder to not buy advance tickets for our first evening. We've been okay sometime (Seine River Cruise, when we arrived to Paris, for example - but that is light, you just sit, and it's only an hour). So yes, we would definitely prefer to take it easy on the first day and night.

Chani, yes, we will look for the earliest entrance to La Sagrada Familia.
I wouldn't mind taking a walking tour of the Barri Gotic, but the rest of the family is quite tired of them. We took so many last year in Italy.
Thank you so much for the Montjuic/Magic Fountain tip.. I need to read up on all that. That's one part that I haven't done enough reading on. Last night, as we were all going through my list and reviewing what interests us the most, no one seemed particularly thrilled with the Magic Fountain. My son, maybe, but that was it.

They weren't too keen on MUHBA/Bunkers either.

What we would all very much like to do, as of now, even though it is a bit of a distance, is the Templo del Sogrado Corazon de Jesus.

One more question:
Since we are five adults, I understand that we may need to take two taxis. I wonder if we should do that or if we should ride the metro. We’re fine with either.

Going with two taxis may be a bit of a hassle, so I would definitely consider using public transportation. The metro is great in Barcelona, easy to use and efficient, the buses within the city are also a good alternative to taxis.

I would look into getting a T-10 card for your family, it can be shared between persons and is entitled to 10 trips on either the metro, buses, and even trams of Barcelona. A single metro ticket goes for €2.20 while the T10 ticket goes for €10.20, which saves you almost €12. One can buy the T-10 at any metro stop and at the airport too.

The metro is easy and fast. Taxis can get bogged down in city traffic.

Carlos and Chani, thank you both. Yes, we'll get a T-10 Card. I agree that taxis can get stuck in traffic. Our Barcelona stay is not very long. Yes, saving money is important, but time is also valuable. We don't want to waste time being stuck in traffic.