After Robin has been knocked unconscious, Batman - who is under Black Widow's mind-altering spell - obediently picks him up and slings him over his shoulders like a sack of potatoes, then follows Black Widow and her goons into their hideout. Once inside the Spider's den, Batman unloads Robin and hands him over to a pair of goons, who carry him away. As Batman is carting around the "unconscious" Robin on his shoulders, Burt Ward can clearly be seen giggling.
As the spiders move up Batman and Robin in the opening, you can see the thin fishing line pulling them upward, connecting to the spiders' abdomen. Around the end before Batman electrifies the spider web, you can clearly see the spider at Robin, is no longer clinging to his body or the web, it's dangling in the air.
When the dummy Grandpa is in the Batman suit, you can see his eyelids fluttering.