Good news and bad news...you can probably (and easily) handle the payment son a $175k house. The bad news - if you settle that debt of o$7K for $2K, it will be marked "Paid/Settled" which is a HUGE red flag. The worse news - because of the blemishes on your record, if you can get approved, you are looking at a conventional mortgage - FHA (which requires only 3.5% as a down payment) requires two full years of unblemished credit - obviously, that clock for you does not start ticking until you pay off both sets of collections accounts. So, going conventional, with the collections accounts and if you make the $2K payment instead of the full $7K, you have the settlement on your record, you may have a tough time getting approved and if you do, the investor is probably going to want 20% down - about $35K. Add in 5% or more for closing costs which is another, say, $8K and you are looking at needing $43K in the bank before you even start making offers.
My suggestion - wait the two years, finish cleaning up your credit, save some more and grab a conventional mortgage when you have $50K to put down plus closing. Why? Because is you go FHA with less than 20%, you have to pay the FHA equivalent of private mortgage insurance (this is required for any loan with less than 20% down). The difference between the FHA PMI and conventional mortgage PMI is that when you hit 78% loan-to-value for a convention mortgage (22% equity), you can get the PMI dropped (saving money), with an FHA loan, you pay PMI until the loan is paid off.
And just to close, there is no fast way to go about this.