I've spoken with my family without the 零 in numbers that are slightly higher (like 6070), and found that many people usually don't put the extra 零 in the number when saying it.

Yes, I guess you could say it is. You can't omit it in Chinese.

I have the same problem here - my logic tells me that you can't possibly get any other number than 6070 when you leave the 零 out. Can someone explain whether it would just sound weird, or whether it would actually cause confusion?

Yes, I've never seen "零“ in real life in China, but I see "〇“ all the time.

No. But you do start using that type of syntax when you get to the 100 thousands. For example, 万 (wàn) is 10,000 and 十万 is 100,000. Also, 亿 yì is 100 million and 十亿 is a billion.