Regarding the question about the Real Presence: Lutherans, definitionally, believe in the Real Presence. This is because every Lutheran accepts the Augsburg Confession, which states in article 10 (quoting from the Kolb-Wengert translation):
X. Concerning the Lord’s Supper
Concerning the Lord’s Supper it is taught that the true body and blood of Christ are truly present under the form of bread and wine in the Lord’s Supper and are distributed and received there. Rejected, therefore, is also the contrary teaching.
If one calls himself “Lutheran” and rejects the Real Presence, he is in no way a Lutheran, even if he should be a member of a Lutheran church body, since “Lutheran” indicates holding to a particular confession.
Disclaimer: I have never been WELS, nor do I know any WELS Lutherans. So, I will say only this about them, and let others fill the rest in: Of the major bodies, WELS is the most conservative.
I have been both LCMS and ELCA. I was married LCMS, and have family who are LCMS (they followed me), and attend an LCMS institution for my Master’s. I am currently, officially, ELCA.
The main differences between them are these:
- The LCMS officially rejects historical criticism and holds to a six-day creation. The ELCA accepts historical criticism and evolution.
- The LCMS practice closed communion, where only members of an LCMS congregation or a sister church with whom the Synod has “Altar and Pulpit Fellowship” may commune at an LCMS altar. The ELCA practices open communion, allowing all the baptized to commune; they also may share clergy with the Episcopalians and several others.
- The LCMS has male-only clergy. The ELCA allows women clergy.
- In the LCMS, each district is headed by a District President who makes no claim to Apostolic Succession. In the ELCA, each synod (confusing, I know) is headed by a bishop who does make claim to Apostolic Succession by way of the Episcopalians.
- The LCMS forbids homosexual unions. The ELCA allows them, though there are still conservatives and moderates who oppose them.
- The LCMS holds to the Book of Concord “because it is a true explanation of Scripture.” The ELCA holds to the Book of Concord “insofar as it is a true explanation of Scripture.”
Lutherans in America are, as a rule, quite congregational in polity. In my town at least, the ELCA church I attend has only three differences between the LCMS church I used to attend: 1) It has three pastors, and one is a woman; 2) the translation for the Divine Service is objectively worse; 3) it uses the Sign of Peace.
The division itself is largely due to historical immigration patterns, mergers, and theological disputes. The LCMS is mostly German, the ELCA has more Scandinavian ancestry. Both are the results of mergers of smaller Synods (the LCMS happened a while ago, while the ELCA came about more recently), each with their own internal debates. In the words of the senior pastor at the ELCA church: “We are often harshest with those with whom we are closest. Our division is that of a sibling rivalry.”