So now that I have had a chance to look at the German birth records from Gelsenkirchen, as well as the translation of two of them, I found out some things about the Linneman family:
First, on every birth record, all family names are listed as Linnemann (not one "n" but two at the end of the name). The passenger list from their arrival in Baltimore in 1904 also lists them as Linnemann. Most US records for the family are spelled Linneman (with a few misspelled versions like Lannemann, Lineman and Lennaman).
The letter from the Rathaus that came with the birth records listed two other children that had died before they left Germany. Rudolf Linnemann was born on 12 February 1893 and died in 1896 and another Rudolf was born in 1899 and died in 1901.
Most of the birth dates on the children's records are the same with their US records with a couple of exceptions. Wilhelm (William) was born on 20 September 1890, but all us records list his birth date as 20 September 1892. Georg (George) has US records that are just two days off from his actual birth date (May 24 instead of May 22). Elisabeth (Mem) was born on 12 June 1897, but used 11 June 1896 as her birth date in the US.
The birth records also listed their middle names, which were never used on any US records and many family members also used American versions and spellings of their names. Elisabeth Maria became Elizabeth, Wilhelm Julius became William, and Georg Adolf became George. Christian Fasel became Christian Linneman and Gerhard Bernard only used Gerhard.
Most helpful was what I have learned about my great-great grandmother Elisabeth, who was know as Barbara Elizabeth in the US. Before Elisabeth married Gerhard Linnemann, she was married to Christan Fasel. Christan was a coal miner who died on 8 June 1887, less than three months before their son Christan was born. The birth records also list Elisabeth's maiden name - Nilkowski. This name sounds like her family was from eastern Germany, closer to Poland.
Lastly, I learned their religions and occupations. Gerhard was a coal miner, which we knew from when they lived in Monessen. Elisabeth was listed as Catholic on Christian's birth records and Evangelical Lutheran on Elisabeth Maria's record. These religions (as well as addresses that are listed on the records) will be very helpful as I search for more information on the Nilkowski and Linnemann families!