If I use Elvish Piper to put Genesis Hydra onto the battlefield by paying 1 forest and tapping it, may I pay an extra 5 forests and reveal the top 5 cards of my library?
No. Putting Genesis Hydra onto the battlefield with Elvish Piper is not the same as casting it (see also your previous question and this thread; review C.R. 601.2), so Genesis Hydra's first ability won't trigger. Moreover, Genesis Hydra's X will be 0 in this situation (you can't choose what X is here), so Genesis Hydra will enter the battlefield with no counters on it (C.R. 107.3e, 107.2; see also this thread and this thread).
Also, you don't "pay 1 forest" or "pay 5 forests", but rather you pay G (one green mana) or GGGGG (five green mana), respectively.
Also, you don't "pay 1 forest" or "pay 5 forests", but rather you pay G (one green mana) or GGGGG (five green mana), respectively.