This thread is about the national day of prayer. If you're going to emphasize the need to focus, focus on that.
Can Lutherans in the U.S. come together for prayer?
That would depend on which Lutherans. The WELS has a unit concept of fellowship, so I wouldn't expect to see them present. But I've been in many situations with other Lutherans and have had no problem with it.
Will LCMS clergy gather to pray with ELCA clergy? What restrictions might be in place?
I've done it several times. No one has ever placed any restrictions on me. Wouldn't know why any would be.
Historically, Walther and the boys had several "free conferences." Each day, those meetings began with Matins and ended with Vespers. This was the practice until the Presdestinarian Controversy at which time the LCMS moved to a unit concept of fellowship. That was one of the issues that the ALPB argued against--and one they wound up winning.
A report (whether true or not, I don't know) was that when the District President of Kansas was to receive an award from Bethany College (an ELCA school,) he was allowed to participate if it wasn't a worship service; specifically, as the story goes, no vestments and no candles.
This whole scenario sounds weird to me. The only way it makes any sense is if he was the commencement speaker and they planned to give him a honorary degree (or something like that). That's the only time anyone who receives an award from a college also participates in the ceremony itself. Otherwise, those being honored sit off to one side and come up when their name is called. I don't know why that would be an issue even if it took place in a service.
About 20 years ago, the ELCA church in Springfield, MA installed a new pastor. I, along with many other Springfield clergy, was invited to attend. I had gotten along quite well with his predecessor and I knew his brother via LTHRN-L. The bishop (who I knew from our New England Lutheran Dialog) asked me if I would mind saying a few words of welcome following his installation. I sat, unvested, but wearing a collar, with the congregation. He asked me to come forward. I welcomed him to Springfield. I said that I prayed for God's blessings on his ministry and I looked forward to working with him wherever I could. The bishop said he appreciated my words; I thanked him for his invitation, which I thought was quite gracious. No one had any problem with my "participation." I'd say the same thing would be true here.