No, it specifically denies establishment.
No, religion has a wider definition than the one you permit - philosophically or otherwise. Your inclusion of "rightly" betrays your subjective bias.
No that is technically true, but only one specific religion prays in this specific way to these specific conceptions (Christianity) and thus, for the State or organise or mandate a 'prayer session' through the Public School system, it is organising and mandating a specific and identifiable religion, and this is a violation of the principle of separation.
Not quite. It can be on the coinage because coinage originates and technically operates as a private enterprise in the USA. Thus there is no violation here on the strict legal terms that coinage is not explicitly a creation of the State.
No doubt.