I just drafted a player in the first round, offered him his original salary he wanted.  He rejected and countered with wanting five more million dollars.  I don't want to pay that.  If I withdraw my current offer to him (his original asking price), will I still receive a compensation pick in the following year's draft?  Or is there something else I need to do?

Just want to make sure that I get the pick in next year's draft.

Thanks

As long as you offered his original asking price and he rejected it, you will get the compensation.

Don't withdraw offers.   He rejected so the money is back in your prospect budget.

Do you only receive compensation picks for the first round? What about supplemental rounds? 

Only for first, second and third round picks.  Not for the supplemental round (between the first and second).

If he rejected it what is there to "withdraw"? As per above, never, ever, withdraw the 1st offer. And remember, in order to increase your offer you must first withdraw the current one.