@Derek Rocco - A couple of things to watch out for (that you may have already included, which I missed):
1. There's a difference between 'Vacancy' and 'Economic Vacancy'. If all 5 units are occupied all of the time, you have 100% Occupancy, and 0% 'Vacancy', right? But if you're looking at a 5 unit building in the Pittsburgh area in the $80k-$90k range (or anywhere in the 12-16 Cap range) I would guess that you're going to have a higher than average 'Economic Vacancy' - i.e. your tenants are going to pay late, under pay, or not pay at all. So you lose that first month's rent, and it takes a month to evict them, meaning you'd be out 1-2 months rent, plus repair costs after eviction (and there will be) plus time to get it re-rented, etc., etc. Thinking along these lines, the current tenants might have only stayed so long because they're being allowed to pay late, skip months, or otherwise cost the owners money - and the owners just don't feel like dealing with the hassle of an eviction to get more bad tenants. Economic Vacancy is a much bigger risk in these types of properties than True Vacancy
2. I'm not sure I understand your reasoning with wanting to use a hard money lender versus a commercial loan on this deal. If you can just barely cover the down payment on it, maybe the smarter move is just to save up for a little while longer, rather than pay the extra interest and closing costs for hard money. Or bring in a more experienced partner for the deal (second set of eyes never hurts on a first deal.) My real concern would be that many commercial lenders won't be interested in this deal, for a couple of reasons. For them, it's a very small dollar amount to lend, so they won't make any money. Add that to the fact that it's a 12ish Cap deal, probably not in a great area, and I think you'll have a hard time finding a commercial lender that wants to take the risk. Apart from that, many times commercial appraisals and inspections are much, much more expensive than residential, and your closing costs are likely to be higher than you expect. So my worry would be, if you take the hard money to buy this, you're going to get stuck in it 10 months from now trying to get refinanced, and wind up "buying time" (to the tune of a couple of points) from your HML.
3. You made an offer, it got rejected - why not make another offer? I know in some markets you "have to bring your highest and best offer" - but I really don't think you're seeing that type of competition for small MF in Pittsburgh (or am I wrong?) Not that long ago I offered $69k on a property that was originally listed at $153k. They were so angry about the offer that they "countered" with a higher than asking price number. After negotiating for several weeks, I was able to buy the property for a huge discount, with another separate parcel of land thrown in for good measure. Get used to hearing "no" - it's just part of the business. (Wait until the first time you're sitting across the table from someone and they completely flip out at your offer. If you want to get that property, you're going to have to "Peel them off the ceiling" - as we used to say in the sales world.) In some markets, this is the wrong approach, but I'm from the school of thought that says "Any first offer that's accepted was too high."
4. The "No Contingencies" statement is becoming more and more common, because there's a trend in Real Estate for buyers to make an offer, have it accepted, and then use a painfully meticulous inspection to try to hammer the seller to reduce the price further. I've seen some absolutely ridiculous things show up on inspections, and have buyers expect money off for them. In my mind, an inspection should cover major things - but if a buyer didn't notice some peeling paint behind a storm window, or old caulk around a bathtub when they walked the house and made an offer, I certainly don't want to give them another $1,000 off to have it painted or re-caulked. So I see this as an attempt to avoid getting nickel and dimed to death. Or there's a major structural issue they're trying to hide. Either-Or. (Have fun figuring out which!)
5. I'd be wary of the thinking that this property is going to be worth twice in 10 years what it is right now. I could be wrong, but I also doubt you're going to see anything trading at a 12-15 Cap right now double in value any time soon.
6. I only see your returns from the calculators - I'd be much more interested in seeing what you put in for costs/expenses. In a 5 unit building with $2200 total rent ($440/unit/month) I would be estimating my expenses to be much higher than the suggested ranges on the BP calculator. And one bad tenant (out of 5) could easily cost you a year's worth of profit in a month - that's always the risk with these types of properties.
7. The Pittsburgh area is one where I would not be at all opposed to buying SFRs right now - especially using the BRRRR Strategy, considering you have a HML already lined up. I think this is an area that's going to grow quite a bit in the next decade (but that's just me.)
Good Luck!