"Are there any ammunition companies left alive today that use the same headstamp as was used in the old west? Did ammunition in the 1870's even have a headstamp at all? What would a period-correct cartridge look like?"
To your first question, believe it or not, yes.
The H on Winchester rimfire ammunition has been in use since the 1880s, IIRC. It's for Benjamin Tyler Henry.
And yes, ammunition in the 1870s could have a head stamp. Not nearly as common as today, though.
I'll try to remember to dig up and photograph the headstamps on some of the older cartridges in my collection.