If a person was sick and did not daven shachris and musaf and now it is time to daven mincha (and there is enough time to daven all the tefillos), what is the proper order to daven? Also, does the person say anything in between the shemoneh esrays, or should he go straight into the next shmoneh esray?
The proper order to daven is to first daven mincha , then musaf, and then the tashlumin for shacharis. The reason is as follows, whenever we daven a tefila to make up for a missed one (called a tefilas tashlumin) it has to be done adjacent but after the tefilah that we are required to daven at this time. Therefore we will always daven the mincha before the tashlumiin of shacharis. Additionally, a person that did not daven mussaf and the time for mincha has already come, should daven mincha first because of the rule of “tadir usheino tadir, tadir kodem”, that when we have two mitzvos to do, we do the more commonly done mitzva first. Therefore you will daven mincha before davening mussaf, and after doing that, you will daven the tashlumin of shacharis 9 this is the opposite of the normal order, which is shacharis musaf mincha!).
As a side point between the shemona esreis, you should say ashrei s a preface to the shemona esrei. To sum up, the order of what you should say in this situation is ashrei, shmona esrei of mincha, ashrei, shmona esrei of mussaf, ashrei, shmona esrei as a makeup for shacharis, and then aleinu.
Shulchan Aruch O:CH 108-2 M:B 108-13, 16,