To answer your first question, technically Spanish American War service was considered any time between 21 Apr. 1898 and 4 July 1902. If he was stationed in the Moro Province of the Phillipines, the date is extended to 1903. So he does fall within the dates to be considered a Span-Am War veteran.
The 3rd Ilinois was a militia regiment, what to day would be the National Guard. He might have had a sponsor or family member already here enlist him in the unit. That would make him more likely to pass the immigration inspectors if he had no money upon arrival, but he had papers showing he had a job and sponsor waiting for him. Or the dates could be slightly off.