I think the HK speakers have badges on them to indicate.
The three ways will have tweeters already so you don't need to upgrade those. I'd forget about the centre dash speaker although it will be powered by the standard amp in the car (Audio GateWay if it has one)!
I've had good results from combining both tweeters and coax in the past but then I like top end clarity and a bit of volume there.
You could start with the 3 way speakers and see how you get on. I once fitted a pair of 6 x 9" co-axs into a MK4 Escort parcel shelf and most people thought the car had a subwoofer - and that was running off a fairly decent Kenwood headunit (probably similar output to your one).
When I bought my W211 (E55), it didn't have HK but the research I did into it suggested it had the same speakers albeit was 150w instead of 250w and was missing the subwoofer and two small speakers either side of it.
I did install a subwoofer as above but the system was very good without that so you might find it's enough to just do the replacement speakers front and rear.
Word to the wise though - spend the most you can afford on the speakers if you can (you don't have to go crazy but I'd consider looking well north of £30) - especially if you intend to keep the car. I did a lot of installs into my own cars in the 90s and early 00s so was used to OEM speakers which were utter shyte. Imagine my surprise when I bought some aftermarket speakers (about £30) and installed them in my 2004 Saab 9-5 Aero around 10 years ago, and I did once side first and compared and the new speakers were utter shyte compared to the OEM speakers in it!
If after all that you find you want more lower end, you can install a slim sub as you state. The problem with those and a saloon (especially a W211) is that it's quite hard to get enough bass into the cabin. So you would have to consider installing one in the cabin somewhere.
Also, you wouldn't be able to drive a subwoofer off you headunit and get any kind of sound from it. The headunit cannot deliver anywhere near enough current to drive a subwoofer. Although a subwoofer is really just a woofer cut off at frequencies below your woofers, lower frequencies require the most current from the source.
You could buy an active subwoofer which has an amp built-in. These can sometimes be driven off high-level inputs or you can buy adapters meaning you can use existing speaker wires which you don't need.
If you have to put stuff in the boot, I'd consider going amp and proper subwoofer (you can buy small 8" subs which enough oomph these days).
The bottom line is there are few quick/cheap routes to success with this gig that will sound any good at any decent volume.
What you don't want to do is slowly upgrade in terms of buying lots of kit and then having to either sell it or take a hit.
So in your boat, I would start with the speakers as you can continue to use these no matter what, and even amplify them if volume is lacking (you will distort / clip them if you drive the too hard with not-enough power). Things like slim subs etc... have a word with yourself first and decide if you are really going to be happy with one.
Then take it from there depending on what you find is lacking.