I think the lever vs. pump question is a good place to start. Get used to the phrase 'I think' here because this is just my observations and opinions.
Winchester et al got there firstest with the mostest, making a solid place in the market with leverguns before a competing pump design. The Winchester 73 in particular had a refined design lineage, with iterations going back to the Hunt, Jennings and Volcanic. By comparison, the Colt Lightning hit the market over 30 years later. It differed in operation in ways that worked against its popularity in the early repeating rifle market.
The lever action arrived on the shotgun market late and did not prove particularly well suited to the task. While the Spencer pump shotgun actually reached the market before Winchester's 1893 and '97, it was the latter that really sold American hunters on the idea of a magazine repeating shotgun. I'm not really a shotgun guy, but I'd guess the 1897's smooth action and straight-line operation were better suited to shooting at birds in flight than anything that came before.
The pump rifle was a more popular option in rimfire than centerfire, beginning its popularity run with the Browning-designed Winchester 1890. It had an advantage over early autoloading .22s like the Winchester 1903 in that it could function with a much wider range of ammunition. Winchester introduced their .22 Winchester Automatic cartridge for this reason. While there are still a few .22 LR pump rifles on the market, I'm pretty sure most of their potential customer base gradually switched over to autoloaders once smokeless rimfire ammo became more consistent and rifle prices dropped.
The next wave of pump-action centerfire rifles were marketed primarily by Remington, starting with a series of John Pederson designs. These are fascinating guns that worked well enough, but they are kind of weird and the mechanics are hard to understand. Check out the animation sequence from this video:
They had a degree of popularity, but didn't really challenge the lever action in the minds of most American sportsmen. You would think they would appeal to the police market that used pump shotguns, but I don't know whether many large police sales ever materialized.
Anyway, Remington has stayed with the pump-action centerfire rifle all through the rest of the 20th century and beyond -- the 7600 was still going when Remington went Chapter 11, and given the demand in some markets (eg Australia) the .223 Remington 7615 might see a limited comeback someday.
The 7615 was a box-magazine pump, a concept that currently has a smallish market as one of the manually-operated black rifle configurations. It's appeal is largely as a legal workaround in restricted markets, but it has the manual repeater's advantage of operating with a wider range of ammunition than an autoloader. It was one of the reasons I bought my Troy PAR -- the action will function the same with just about any .223/5.56 handload. Not much of a advantage I grant, but I also got a pistol grip and folding stock out of the deal.