1. That lender screwed up. There isn't a loan product I know of for residential mortgages that will allow you to put less than 20% down on an investment multi unit. In fact, most will not let do this type of deal without 25% down. I have personally financed my last two had to put down 25% for both to get a Fannie Loan.
2. PMI will not help you because PMI companies are not going to issue you PMI on an investment property unless it is a single unit. You can find PMI on a single family unit or condo, 15% down, but it is expensive.
3. I am curious about this myself and why any seller would want to do this unless they were making a very serious return on doing this for you. That is the whole reason they are trying to sell the property typically.
4. Banks do not need to wait any period of time between loans on different properties, if that is what you are asking. If you are asking how long they need to wait before getting cash out, that is typically six months. Unless you bought the property for cash - then you can do something called delayed financing to pull your money out in less than six months. You are also not maxed at eight loans. You are maxed at a total of ten financed properties and that typically includes your primary residence.
Not to continue to call out any inaccuracies in the other answers but you cannot do FHA on an investment property. You must live in it. Also, residential mortgages only go to four units, not five. Anything more than four units and you are into commercial/portfolio territory. Not residential.