Is there such a thing as annulment of marriage in the Lutheran church? What are the grounds for it? Also, while I know what the Bible says about divorce, provided both partners in the marriage stay faithful to their vow and don't get involved with another person romantically, is legal separation permissible and acceptable if the main motive is to resolve the issues that are causing problems in the marriage and the other goal is to get back together as soon as possible?
In the Lutheran Church there is no provision for an ecclesiastical or theological annulment of marriage. This means that we do not recognize any Biblical basis for declaring that a marriage that took place never really existed, theologically speaking. Roman Catholic marriage annulments are based on church laws and decrees, not the Bible. We do well, however, to distinguish a “theological” annulment from a legal, civil annulment. A civil annulment is part of the laws of the state in which you live and those rules are distinct and clear. A civil annulment declares that while a marriage took place, it was illegal and will not be recognized as valid. If an already married person gets married to another spouse — that is, when bigamy results, — the second marriage is declared illegal and void.
A formal separation, whether established legally or merely agreed upon by the partners, is a legitimate action as described and with the purposes stated by you. 1 Corinthians 7:5 establishes that kind of pattern, although with a distinctly different purpose (to devote yourselves to prayer) and also with the reminder that separations may bring with them other pressures (unfulfilled sexual urges). To seek a formal separation by mutual consent as described by you cannot be classified as wrong. But this should be done carefully, cautiously, with ongoing appraisals of the situation, and renewed resolve to deal with issues in anticipation of reunion. Pastoral counseling and perhaps other professional supervision is advised.