63/37 is euctetic, 60/40 isn't. Euctetic means solder transitions between liquid and solid almost in an instant.
63/37 is better as it "snaps" to solid state at an instant when 183c is reached, but 60/40 stays semi-liquid / liquid for a longer temperature range (around 180c) making it easier for beginners or people with low quality irons.
63/37 is often recommended to be used in environments where there's lots of vibrations or other noise or issues that could affect how the joint solidified. For example, on a ship where the ship engines vibrate all the time you wouldn't want a tall capacitor's leads to vibrate and move around while the solder becomes solid, that can result in a weak/bad solder joint .. with 63/37 you can apply solder and hold part for a second or so until solder cools down and snaps to solid.
There's also other formulations which are euctetic, for example I think - from memory, but fairly sure it is euctetic - 63/36/2 Ag - they use 2% silver. The added silver helped when a lot of surface mounted components had silver in their terminals.
Don't know what "glow core" means.
There's rosin core flux, there's RMA flux (rosin mildly activated) which has some additives which makes the flux more active but rarely it's required to clean the flux from the board and there's RA flux (rosin activated) which is rosin with extra stuff, and depending on formula it's sometimes recommended to clean the flux off with isopropyl alcohol or other solvents because otherwise over time, the flux could still work and affect the circuit.
There's no-clean fluxes which as the name says don't have to be cleaned off the board and usually leave very little residue. A lot of no-clean fluxes are as strong as rosin or RMA fluxes.
Then there's organic and water soluble fluxes which you should avoid - they're stronger fluxes but the smoke is also worse for your lungs and those water soluble fluxes are not quite as easy to clean off the boards as the name says, you can't just wipe the board with a wet towel. You're supposed to be more thorough, and even use distilled water, make sure there's no flux under chips... it's a pain in the ass.
Some solders have a single core of flux along the length of the wire, other solders have several cores (multicore has 5 for example) - the flux is supposed to liquify faster than the actual solder and drop on the parts you want to solder and remove/attack the oxides and corrosion on the surface of the metals, making it possible to transfer heat better and for the actual solder to form a chemical reaction with the pads /other solder on the leads. Liquid/gel flux on the components leads/pads that you want to solder always helps, as sometimes the flux inside the solder burns too fast or is too low amount.