My favourite Christmas market by far is the Feria de Santa Lucia, a lot of good memories there. It's oldest one in Barcelona from the 1700s, it's held from November 30 to December 23, and comprises of around 100 stands. The market is set up in front of the Cathedral of Barcelona, in the Gothic Quarter, so it also has a unique setting.
The biggest draws are the nativity scenes/figures (typical across Spain) and the Tio de Nadal figures (very Catalan tradition), those happy faced Christmas logs you mentioned. Children take good care of the log, keeping it warm with a blanket and feeding it, so that on Christmas Day, it defecates small candies and presents. One actually bashes this poor log with sticks, at the accompaniment of traditional songs, so that it poops out better presents or candies. One of our many quirky Catalan traditions lol!
Another local Christmas treat is what we have is called Turron – a nougat-like sweet usually mixed in with toasted almonds or even chocolate. I would be on the look out to try some. For sure, you can find a great selection of different turrones at El Corte Ingles.
If it gets a tad too cold, nothing brings back the warmth like a cup of typical Spanish Hot Chocolate, one can find many hot chocolate establishments around Barcelona. Valor Barcelona is a pretty good chain.