I would not turn the bars over but change them to a more upright bar, along with the stem. As for tires that is simply a matter of what fits in your frame.
People have been turning old road bikes in cross bikes for years. back in the early '90s we used to do several "hybrid Conversions" a week to old low end 10 speeds.
However depending on just what type of trail riding you want to do, you might be better off passing this along and buying a used better quality hybrid of all ChroMoly construction or a real cyclocross bike.
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“One morning you wake up, the girl is gone, the bikes are gone, all that's left behind is a pair of old tires and a tube of tubular glue, all squeezed out"
Sugar "Kane" Kowalczyk