We will be in Barcelona the last 7 days of October. We purchased tickets for La Sagrada Familia on the 30th to make sure we got the first english tour of the day. Do we need to book any other places that late in the season or can we wait until we get there?
You may be able to wait, not because of the timing of your trip (Barcelona will still be very busy in October), but because you will be in town for a week. That provides a lot of scheduling flexibility. However, you may find all the first-time-slot tickets sold out if you just look for tickets a day or two ahead of time. I think trying to pick up online tickets early on the day of visit might not go well.
Do not show up at any of these places without a ticket already purchased: Parc Guell, Casa Mila/La Pedrera, Casa Batllo, Picasso Museum, Palau de la Musica Catalana.
I would absolutely book tickets in advance. If you're sure you're going to go, and can take some time to plan out your trip in advance, it saves you so much time having the tickets in advance.
Plus, when you plan it out ahead of time to buy the tickets in advance, it takes the guesswork out of the day to day when you're there. We did this when we went in September a year ago, and loved having our whole trip scheduled out and all tickets in hand. It gave us more time to wander around the area prior to each excursion, take in nice meals, and not wait in line wasting our time.
My advice would be one activity/site per day, otherwise it's too stressful and timing is too hard to predict.