to spread something out flat over a surface: to lay cloth, branches, or bedding down so that it covers what is underneath.
The picture behind this word is simple and physical. Something soft and flat gets laid out across a surface until the surface is covered. What changes from scene to scene is only what gets spread, and why.
Outside Jerusalem the crowds pull off their cloaks and cut leafy branches and lay them across the road, turning the road into a carpet and the arrival into a welcome fit for a king. Luke tells the same moment with a related verb built on this one (ὑποστρώννυμι), which spells out that the cloaks went down underneath him as he rode over them.
The other two uses move indoors, and both are ordinary household language. A room that has been "spread" is a dining room with its couches and their coverings already laid out, so English Bibles call the upper room "furnished". And when Peter tells Aeneas to get up and spread for himself, he is telling a man who has not risen from his mat in eight years to make his own bed. Nearby, Mark and Luke both reach for a broader word for getting ready (ἑτοιμάζω); this verb is the narrower one, naming the actual laying out.
Nothing in the word itself is religious. Its weight in each scene comes from who is doing the spreading, and for whom.