Is gold more valuable when it's minted vice powder or chunk form?  Also, can you mint your own coin?

I'm talking same purity and same weight.

You pay a little extra for coin because they are of known, usually marked, purity and weight and produced under government standards with a recognizable design. The thinking is that they are harder to fake or adulterate and easier to trade with vs needing a scale to weigh out the amount for a transaction. I keep a few coins on hand for SHTF purposes because they are easy to carry/handle/use and I think a gold coin has a dazzle factor that's beneficial in the negotiation stage of a transaction. US Gold Eagles for me, although if I were traveling internationally, among countries where there may be some hostility to Americans, I'd be perfectly happy withh Canadian Maple Leafs.

Now, if you are talking actual investing, most investors don't hold any actual gold, they hold a paper title to some gold, probably bullion, that sits in some company's safe. It's weight and purity will be certified by the company and that certification is as good as the company's word, for better or worse. Whether you can go to the company office and ask for a brick from "your" pile of gold will be a matter of contract.

Whether you can mint your own coin... in the USA, I'd say the answer is no if you actually mean  "mint" since our mint is a government entity operating under government rules and standards to produce legal tender. If you just want to strike a coin for shits & giggles or to consolidate gold dust into a convenient chunk that won't leak through the seams of your pocket, sure. People may be skeptical if you chisel  "Bank of Hardcorps 1 Beeelion Dollars!" across the bottom but you are probably more trustworthy than the Chinese so hey, who's stopping you? Your coin will not be legal tender though.

If you are actually in a position to amass enough gold to stack on a pallet, rather than carry in a pocket, then I hate you. Large holders prefer bullion (bricks) for ease of storage/transport.

Take any of the options, melt it into a lump, it's now all of the same value.