Do you need the double bed to get married + kids in story of seasons?

Your house needs to be upgraded to include a double bed, yes.

The Big Farm House is the upgrade you need to be at. The blueprint for the house upgrade will be at Gunther's shop for 100,000 G after you build the Upgraded House (or 70,000 G if you are playing in Seedling mode). The Big Farm House requires 60 Lumber, 60 Stone, 10 Glass Stone, and 7 Emerald.

The house comes with a double bed inside of it. You can change the style of the double bed if you want, but you don't need to have a specific double bed style to get married.

source and more reading for marriage here

As for your second question, yes. There are kids in the game, The type of wedding you choose to have determines how quickly they show up (more expensive wedding = quicker kids). The kids are always twins(1 boy, 1 girl). They don't do much(like help around the farm... lazy bums), but there is a few fun events that they trigger.

and source and more reading about kids here