No, collections agencies cannot actually have the credit report agencies delete a late or delinquent payment. They might lie and say that, but if it is a legitimate debt, the best they can do is indicate that the charge was paid in full, and when. Often, that is enough.
A medical bill for an amount of approximately $100 CAN affect your credit score, for purposes of getting a credit card. I know this because I had a similar situation to you-- I moved across the country and never received that piece of forwarded mail containing my doctor's bill co-pay.
I only found out about it because I was decided after applying for a new credit card, solely for that reason. By then, the bill was 5 months late, so that it showed as seriously delinquent (over 90 days) on my credit report (FICO/Experian) and had been referred to a collections agency.
I promptly contacted the collections agency, paid the $80.00 debt and asked if it could be removed from my credit report. The credit reporting agency said that they would record it as paid in full but that the lateness couldn't be removed. I called the original physician who had issued the bill; they couldn't help.
I applied again for a credit card about a year later. I was denied again, due to having a seriously delinquent past account, that same $80, even though I had paid in full. It will take between five and seven years for your medical bill to drop off your credit report, once you have paid it.
Different criteria are used for secured borrowing such as a mortgage or auto loan. If paid, your medical bill probably won't have any negative impact on your ability to get a mortgage in a few years, as that is well-collateralized by the house. Regardless, pay the medical bill now, get a receipt and a full statement from the collections agency that the bill has been paid IN FULL.