When foot archers had to fight close up, what do they do with their bows? Would they throw them away or did they have some where to hold it so it wouldnt get in the way? Likewise how did heavy horse archers carry both a lance and a bow?

They’d usually drop them at their feet, because longbows are incredibly awkward to carry –  and stringing it across one’s back as in the movies is a great way to get yourself snagged on something.

Horse archers tended to have much smaller bows, and holsters that hang off the saddle.  

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As for how they transported the lances, see here