Learn Italian: Ho trenta euro. - I have thirty euros.

No, cardinal numbers from two onwards don't change based on gender. You use trenta (thirty) for both masculine words like euro and feminine words like penne (pens). The only exception you learned earlier is uno, which changes to una for feminine nouns.

Why doesn't the word euro get a plural ending like the English word "euros"?

In Italian, the word euro is invariable, meaning its spelling never changes to show plural. You say un euro for one, and trenta euro for thirty.

Where is the word for "I" in this sentence?

In Italian, subject pronouns like io (I) are usually left out. The verb form ho (I have) already makes it completely clear who is doing the action.

Should I pronounce the "h" at the beginning of ho?

No, the letter "h" is completely silent in Italian. You pronounce ho exactly like the vowel "o". The "h" is just there in the spelling so you don't confuse it with the word o, which means "or".