Which part of Caiscais are you staying in? It's been about three years since I was there but, if I remember, the nicest beach is the third one along the way to Estoril (called Praia de Conceicao). One day we were there someone was taking their ferret down for a dip and she allowed small child to 'help' walk it down to the sea on the lead.
There was a great old-fashioned carousel at Jardim de Visconde da Luz (about a ten minute toddler pace inland from the town beach, Praia da Ribeira). There's also a nice big park (Parque Carmona), with lots of birds wandering (and resulting exotic feathers to collect) and a playground or two. We didn't find any restaurant that wasn't child friendly but both Frangos (just opposite the carousel square) and the Esplanada Santa Maria (walk out along the coast towards the lighthouse, near the Museu Guimares/Parque Carmona) both went the extra mile (which is saying something in Portugal as they are great with children anyway).
It's worth heading into Lisbon for a day too - the train is v.straightforward. It's also easy to get to Caiscais from the airport via underground/train, even with a toddler in tow.
We didn't bother with a pushchair, but mine was a little bit older (3) at the time . There were lots of coffee/pasteis de nata stops though! It is hilly, but the tram & underground are easy to use, so if you know which ones run to places you want to visit, just in case, then it should be fine.
The Pena Palace is stunning, but a bit of a hike uphill. Small child would definitely require help and yes, not sure if it would hold their interest. We didn't get a chance really to look at much else due to the time it took to get around Pena.
We're in lisbon at the moment! We went to cascais yesterday and would recommend flamingo restaurant for good food and friendly service. The oven baked fish Portuguese style was delicious.
As for buggies in lisbon - try to avoid! We've got ours for nearly 3yo dd (and occasionally her 5yo brother...) but it is a pain. Saying that, the tram isn't so reliable so we've had to walk a lot too, and lots of walking to get around metros etc.
Pack a parasol if you can as there's no shade on the beach. And it is getting pretty hot here at the moment. 32 degrees at one point and hotter as the day goes on.