A reference entity (the debtor that might have a credit event) does not have any currency denomination.
A reference entity might have many outstanding debt instruments. Each instrument is denominated in some currency. You need to choose one debt instrument as the reference obligation for the CDS contract.
You choose the currency in which the CDS contract is denominated: USD, EUR, JPY, etc. This currency is used for all the cash flows of the CDS: the upfront fee, the quarterly running spread, the (notional - recovery) if there is a credit event.
When the parties don't want to physically send over the CDS currency, it is common practice to observe its USD exchange rate 2 business days before the cash flow, and to send over the corresponding amount of USD. (It's like embedding a non-delivery forward into every cash flow.) You don't need it for EUR or JPY, but you would do it for a CDS denominated in BRL, for example.
The CDS contract does not have to be denominated in the same currency as the reference obligation, although in practice it usually is.
The choice of the reference entity matters most, obviously. The choice of the CDS currency matters somewhat. But the choice of the reference obligation matters very little (as long as they're pari passu).
If your booking system is linked to IHS Markit Reference Entity Database (RED https://ihsmarkit.com/products/red-cds.html ; REDL on Bloomberg), then the reference entity's 6-character RED code goes on the term sheet; and the database also contains preferred reference obligations for various combinations of reference entities and currencies. So if you're trying to execute a EUR-denominated CDS, and the RED database has a preferred ref ob for a EUR bond, then you'd probably use it. If there is no EUR bond, but there is a USD bond, then you can use that. It doesn't matter. If there is no preferred ref ob in the database, then you need to figure out what you want to use as the ref ob for your trade, but this choice really does not matter as long is it is pari passu with the debt tier that you want to reference.
The ref ob is somewhat of a relic from the earlier days of CDS trading, when after the credit event, the protection buyer could physically deliver a defaulted bond (the ref ob or parri passu) and receive the full notional. No one does physical settlement anymore. Today, an auction determines the recovery $R$, and the protection seller pays $(1-R)\times$ notional in CDS currency (cash settlement).
If a reference entity usually has CDSs in some currency (USD for US names, EUR for European ones, JPY for Japanese ones) and you decide to trade in another currency, then it might make a difference in the price of the CDS for three reasons:
the interest rates might differ between the two currencies. This only makes a material difference if the IR are very different, e.g. USD v. BRL.
technical reasons - supply and demand. E.g. someone might be trying to construct a EUR-denominated synthetic CDO and trading single-name CDS in EUR, while not moving the same names in USD.
devaluation on default. Suppose you buy CDS protection in MXN on Pemex (referencing any hard-curency external-law debt, doesn't matter which specific ref ob). You'd pay less because you'd expect that if PEMEX defaults, then MXN would devalue. Suppose, conversely, that you buy CDS protection on Italy sovereign denominated in USD, rather than EUR. (I mean the CDS currency. The ref ob and its currency don't matter.) You'd pay more for USD protection because the EUR would devalue of Italy defaults.