Most professional bartenders I've seen wouldn't normally pair a Hawthorne strainer with a mixing glass. A julep strainer goes with a mixing glass and a Hawthorne strainer is used with metal containers (e.g., the steel half of a Boston shaker).
SmithyNZI have all of the above (from this manufacturer) and tend to grab a superior variation of what you call a Hawthorne strainer regardless of the container I'm using. If julep strainers were worth a shit, no one would bother with Hawthornes and vice versa. By the way, while the weighted stir spoon is indispensable, mixing glasses are overrated, especially for Martinis (which as everyone knows, should never be stirred). My advice, save a ton and buy these items at Target or Amazon for 1/3 the cost of this package deal.
RayFIt's perfectly okay to stir a martini. If you stir it about 50 times, you get a good chill and a little dilution (which you want) without risking ice bits in your cocktail. If you don't shake carefully enough (most home bartenders don't) you run the risk of over-dilution and bits of ice in your drink if you don't strain well enough. James Bond doesn't know everything.
CheesecakeBond doesn't, but I do. Either method requires pouring through a strainer (preferably not the one on offer here) which eliminates the the ice chip problem. I have nothing against dilution (it's part of the plan), but the objective is to get the drink as cold as possible as quickly as possible. String is an option, but never a preference.
RayFMost cocktail aficionados and bartenders would disagree that shaking a dry martini is preferred over stirring. It's your choice how you have yours though - there's absolutely nothing wrong with that!
SmithyNZI suspect when you use the term "cocktail aficionados" you're referring to people under the age of thirty, which means they've achieved that dubious status in just eight short legal drinking years. Further, how you can speak for most of them is beyond me, but bear in mind you're all at an impressionable stage in life. Nothing particularly wrong with that, but do leave room to change your mind down the line.
RayFI think you'll find a quick google search will answer all your questions, Ray. Both for whether a Dry Martini should be ideally shaken or stirred, and also when to use a hawthorne strainer vs a julep strainer. I think I'm done here. Peace to you.