Great question! In each of those sentences, it would be normal to say them with "at" or without "at." Both versions work in a casual conversation. For number 2, you technically should use "at." However, in each of the situations, it just depends if you want to be more formal or informal in your conversation. Use "at" to be more formal. Don't use "at" if you want to be more informal (among friends).

The sequence of sentence 1 is a bit awkward. The sequences normally used ; Formal Yesterday I stayed at home all day. Or Casual Yesterday I stayed home all day. Same goes for sentence 2, (yesterday) will normally go first in the sequence. Yesterday I was home Yesterday I was at home (Depending on the situation you'd use one or the other ) 3rd sentence I'm home (already emphasis where you are at the moment so "now" can be left out) Depending on the situation "I'm at home" can be used. I'll try to explain a little. If you have just arrived home from work, you would say, "I'm home !" But NOT "I'm at home" If someone asks your location "Where are you? " Here you could use either I'm home or I'm at home