In spite of the fact that you leaked information very economically, it is possible to provide “any information”. The clue is the particle “von”, because it allows to identify the right Rimschas.
The address book 1904 for Riga informs about two von Rimschas.
1. M. von Rimscha, house owner.
2. Rob. von Rimscha, Dr. med.
Similar address book for 1913 informs only about R.v.Rimscha
In the Russia Empire of that time not everybody could vote for the municipal bodies. It was possible, for example, for those who had a real property in the city. The lists of voters were printed, and I have such list for Riga 1890 that informs about a house owner Michael Hieronymus von Rimscha.
Rather surprisingly the information about this M.H.v.Rimscha can be found in the Internet. You are advised to google up “Michael Hieronymus von Rimscha”. The Internet informs even about his children.
Marie Caroline von Rimscha, b. 10. 10. 1860
Michael Hieronymus Antonow Rimscha, b. 12.07.1862
Robert Hermann Wilhelm von Rimscha, b. 24.06.1864
Gerhard Heinrich von Rimscha, geb. 30.06.1866
I have another list of voters for 1909, and this list comprises some information about these children and their spouses. They were co-owners of some enterprises. It can be supposed that this was the generation of the parents of Friedrich von Rimscha. As the first name of Friedrich’s father remains unknown and no additional information even on the level of family legends was provided, it is impossible to decide who was Friedrich’s father.
In any case the information from the Internet site mentioned above supports the idea that in Riga of that time only one family of von Rimscha lived. In case you do not understand the German text of this webpage, it tells that Michal Heronym Rymsza (b. 1827) migrated to Riga from Poland and started all paperwork to recognize his nobility officially. He succeeded and was included in the registers of Baltic nobility. Simultaneously he changed his Polish name Rymsza to von Rimscha. The spelling was changed; the pronunciation remained practically the same. It is difficult to imagine that many Rimschas went from Poland or Lithuanian provinces and successfully registered themselves in the Nobility lists, this is why I think that all von Rimschas in Riga were descended from Michael Hieronymus.
In 1939 together with other Germans all von Rimschas left Latvia, totally the list of resettlers contains 13 persons of this name. A look on this list allows to agree with the idea that they were descendants of Michael Hieronymus. I understand that you have good information about this period because you asked nothing about it.
The googling of Michael Hieronymus provides two sites, the first one was discussed above, the second one presents his photograph of 1863 with additional information that the source of this photograph is the family. My guess, just a pure guess, is that this is the family of Hans von Rimscha (1899-1987) who was born in Riga as a son of the above mentioned Dr. med. Robert von Rimscha, emigrated in 1939 together with his wife and two sons and later was a history professor in Erlangen University. You may try to contact this University and ask them about information about the contacts with the descendants of the professor.
Hope it helps