I'll be the contrarian here and suggest that if the engine instruments are part of your regular scan, there isn't much to be gained by sitting and staring intently at the oil pressure gauge for a full 30 seconds after start. There's definitely a time and a place for that (basically all turbine engines) but I just don't think starting the mighty Lycoming O-235 on a warm day makes the cut. After start and good oil pressure, take a mental snapshot of your oil pressure gauge every few seconds during your instrument scan and look for trends.
I went back and looked and here's what Cessna has to say in the 1979 C152 POH (Normal Procedures, page 4-11):
It goes on to note that there are more specific procedures for cold weather starting which are noted in another section.
Back to OP's question #2, I'd like to point out that nowhere in the quoted C152 POH normal procedure above does it say exactly 30 seconds. I fear like so many things in aviation, a CFI read the "within 30 seconds", paraphrased it to a student who held onto it forever due to the law of primacy, who then became a CFI and taught it to their
students, and on and on...
The reason I listed off so many of those engine parts in question #3 was simply because seeing those physical parts in action and understanding how that engine fits together and which parts are bathed in oil and why will be way more intuitive than regurgitating a canned answer like "because the engine needs oil pressure to work".
I know a lot of pilots' understandings of engines boils down to a black box that makes thrust and noise when you feed it money, but for the engineering and gearhead nerds among us they are really damn cool. On a personal note, I'm extremely grateful I got the opportunity to apprentice as an aircraft mechanic after high school (and kick myself daily for not finishing the A&P rating) because it both allowed me to pay my way through my flying lessons and gave me a hands on understanding of aircraft structures, systems and powerplants that pushed me to go back to school and become an aerospace engineer.
To that end, I know most pilots don't care to dig into systems stuff beyond the basic requirements, but I challenge you to try and find the opportunity to wander into a maintenance hangar sometime and look at all the "stuff" on your airplane taken apart and how it all fits together. It may challenge preconceptions of things you thought you knew, and you'll almost certainly walk away with a better understanding of the machine you're manipulating.
Since I can't provide you that experience through a web forum, I've attached two videos - a beginner one and an advanced/nerdy one. They both feature a Lycoming IO-360, which is fuel injected and makes more horsepower but has the same number of cylinders and a very similar geometry to an O-235.
1. The beginner video is made by Embry-Riddle, and has a very well done computer animation of the engine operating, with cutaways of individual components (14:34):
2. The advanced/engineering nerd version is "Skywardtech: Rebuilding Small Aircraft Engines (IO-360)". Just a dude from the 80s teaching you how to overhaul an IO-360 from start to finish in 36 minutes, including the infamous silk thread gasket at 4:33. I love this kind of stuff, and although I don't expect you to feel the same I hope a couple of my nerd brethren out there get a kick out of it too.