Hello "msulkes" and welcome to the group. Here's a little info relating to your "Hamilton-River" question.
The Hamilton bus line initially looped in the vicinity of Hamilton and Six Mile Road when it was converted over to buses in April 1947 (actually looping the block via Pontchartrain, Whitmore, Third, and Merton). After October 18, 1947, the line was extended northward via Pontchartrain and Seven Mile Roads, and has followed that basic route since.
The "River" destination you mentioned refers to the Hamilton line's downtown terminus at the foot of Woodward Ave and Atwater Street, which it continued to follow until Woodward was closed south of Jefferson in 1953. Beginning April 1, 1953, the route looped downtown via Woodward, Larned, Randolph, Jefferson and Woodward.
The 1951 Hamilton Coach route was described by the DSR as follows:
Southbound—Grand River and Lahser via Grand River, Redford, Lahser, 7 Mile Road, Pontchartrain Drive, Hamilton, Milwaukee, Third, Henry, Woodward to Atwater (River).
Northbound—Woodward at Atwater (River) via Woodward, Temple, Second, Seward, Hamilton, Pontchartrain Drive, 7 Mile Road, Lahser to Grand River.
Note—Starting at 7:40 P.M. to 5:15 A.M. on Weekdays, Saturdays, Sundays and Holidays, the Northbound Coaches Only will operate from Atwater via Woodward, Witherell, Adams, Grand River, Second, Ledyard, Second to Temple, then regular route.
BTW–Riding the Hamilton bus during that time (1949–early 1950s) was probably quite disruptive, considering the John C. Lodge Expressway was under construction and pretty much wiped-out much of Hamilton from Clairmount to Grand River.