But… how specific do I have to be?
The closer the better but, unless you have access to an exact bullet-case-primer-powder combination, you'll have to do at least some interpolating. Try to stay with the same weight, hardness and profile, at least. Remember that seating depth is the real measure and cartridge overall average length (tip-to-base) is just a convenient way of approximating seating depth. Also remember that the real point of measure is the ogive - the point on the bullet where the driving band ends and the slope of the profile begins - not the tip. Again, measuring from the tip of the bullet is very imprecise but it's also very convenient.
Will just about any commercially available lead bullet do, that’s close? For example, in this case, 150 gr bullet? Do I have to worry about what alloy they are? Do I need gas checks?
Yes, yes, and yes. Maybe.
Yes, you can typically find comparable data for just about any commercially available bullet. Some manufacturers tell you to cross-check similar profile bullet data while others give you specific references. That's where bullet choice comes in: do you want specific data or a reference to comparable data? Up to you.
Yes, most any 150gr. bullet with a similar profile and seating depth will have good STARTING POINT data for you to work with. Never jump to MAX or even median loading data. Always start low and work up.
Yes, alloy matters a great deal but, more than that, the composition of the alloy's hardening components is more important than the actual "Brinell Hardness Number;" the percentages of lead to tin and/or antimony can determine how well an alloy performs in a rifle barrel more than just how "hard" it was when cast. Look for balanced alloys - equal-ish proportions of tin-antimony-lead - and castings made for their application - wax lubed bullets with lube grooves, for example. What you're looking for is an alloy hard enough not to skate when pushed through the rifling which is soft enough not to shatter due to rotational inertial forces on leaving the barrel.
Maybe you "need" a gas check to prevent gas cutting and maybe you don't. Some of the new plastic coatings (Hi-Tek) replace lubes completely and lend some resistance to skating to their cores while others are just wax by another name (let's all recall waxes are polymers). The purpose of a gas check is to prevent hot gases from getting past the bullet's base into the gap between the driving band and the bore/grooves. If the bullet/mold manufacturer recommends a gas check, it's probably not a bad idea to use one; if not, then it's probably not crucial. Experiment and note your results.