We don’t daven Tefilos Nedava Bizman Hazeh.
Yungerman: That’s a Chumra from a T’shuvas Harashba that we hold like if possible. You should only start a Tashlumin right after you finish your Shmoneh Esrei. But I don’t think you have to wait only Hiluch Arba Amos. Taking a minute to prepare yourself should be fine. His only requirement is that you still have to be Assuk in that Tefillah.
A Meizid cannot Daven Tashlumin but a Shogeg can. It doesn’t have to be a pure Ones. In fact, we are pretty lenient by this as to what an Ones is.
When Davening Tashlumin one should be very careful to have in mind that the first Shmoneh Esrei is their current Chiyuv and the second is the Tashlumin. Both Shmoneh Esreis are identical though. (i.e. if you missed Minchah Friday afternoon you Daven 2 Shabbos Ma’arivs.) The one exception to this is Havdalah. If you forget Shabbos Minchah then you only say Atah Chonantanu in the <i>first</i> Shmoneh Esrei.