Last past Sunday an Australian language exchange partner showed me an Australian plastic currency, which somewhat surprised me. Last Sunday an Australian language exchange partner showed me an Australian plastic currency, which somewhat surprised me.

You don't need both last and past; additionally past sounds much more natural. Also, adding "me" after "showed" makes it complete, as you need a subject.

There is no doubt that there should be existing the plastic currency existing such as an Australian money or other currencies. There is no doubt that there should be plastic currency existing such as Australian money or other currencies.

I guess that I have fixed thinking about the currency, like the currencyat it should be the paper. I guess that I have fixed thinking about currency, like that it should be paper.

You don't need articles if you're talking about currency in general. Also, adding a comma after your first clause makes the sentence easier to understand.

When I took a look at the plastic currency at first, it's was beautiful and really cool. When I took a look at the plastic currency at first, it was beautiful and really cool.

You want to keep your tenses consistent here, as they're both past actions/events.

I'm not sure, but I guess we had better use the paper currency rather than the plastic currency in aspects of the security. I'm not sure, but I guess we had better use the paper currency rather than the plastic currency in aspects of security.

Again, you don't need to use "the" in front of every noun or it sounds off, as you're talking about a general concept. :)

It seems to me that the plastic currency can be forged much more easierly than paper currency, although I'm not sure ofat all sure about the plastic currency's security at all. It seems to me that plastic currency can be forged much more easily than paper currency, although I'm not at all sure about the plastic currency's security.

It sounds more natural if you put phrases like "at all" in front of the noun you're modifying.