Your type of separation from the U.S. military (other than honorable) should not effect your immigration status in any way. I am sorry that service in our armed forces did not work out for you. USCIS should not consider your service and discharge when determining whether or not to renew your green card. That is, unless you were discharged for committing a criminal offense, which would also be a ground of removability. If that is the case, you should consult an experienced immigration attorney to assist you in this matter.
Generally speaking this should not be an issue. However there are many other reasons your case may be denied so it is best to hire an immigration attorney to assist you.
Alexus P. Sham-The above information is only general in nature and does not constitute legal advice. It does not create an attorney-client relationship.
The OTH discharge MAY affect both your residence status and/or an application for naturalization; it all depends on the reason for the OTH discharge. I would strongly suggest that you consult with an experienced immigration attorney for a full review and advice.
This answer is of a general nature and should not be relied upon as final, nor is it intended as legal advice. A consultation with an experienced attorney is always the best way to go.
No - the fact that you have an honorable discharge does not affect you renewing your green card. You may also want to consider filing for your naturalization at this point.