According to some info that I have been reading, a hospital can turn me over to collections even if I am paying an agreed upon amount faithfully ever month.
If I continue to pay the collection agency every month , will this show up adversely on my credit report?

Yes, it is going to show if you where ever late on your payments. FYI those medical bills reailly hurt your credit score believe me I know from exp.

If you have an agreed to paymnet arrangement and that agreement is in writing as to what they will accept, they should not be able to turn it over as not being paid. The collection agency can not require any additoinal moneys.

Even if you have a payment plan they are still going to report it that you are paying it. And no they can't ask for more money.