I highly doubt a screw was already inside the tire, especially a rusted one. Probably just went through while riding.
You shouldn't go riding without a spare tube and inflator of some kind (mini pump or CO2 are the best options).
The slime tubes are great, and yes its fine that green stuff came out. Thats the sealant, and if some of it got into the valve it's going to come out. There is plenty in the tube so don't worry about the sealing properties being lost.