There are a lot of things to consider with this, especially if the two speakers are not identical. But let's assume that they are...
Theoretically, if you double the wattage then you'll get +6dB more sound level out. This is true (theoretically) regardless of if you take a single speaker from 200 to 400 watts, or go from one to two 200 watt speakers. In both cases the wattage doubles, so the sound level will rise +6dB.
In theory, theory matches practice. In practice, it doesn't.
Things are rarely so simple, and this is doubly so when it comes to sound. When you go from 1 to 2 speakers, you can get strange effects from the two speakers interfering with each other. Placement and orientation of the speakers can have a huge impact of this. In some cases, certain frequencies can completely cancel out while other frequencies will rise as much as +6dB. This is called "comb filtering". There are other related effects that could be a good thing (like a line array) or a bad thing, creating lots of really bad sound.
There is too much here to go into, but suffice it to say that if you want (effectively) a single speaker at 400 watts then that's what you should go with and not two 200 watt speakers. If you don't have that as an option, then I suggest that you place the speakers in such a way as the speaker coverage patterns don't overlap.
There is one thing that you didn't ask about, but I suspect you need to know, is how to calculate the apparent sound level given the wattage and speaker. Knowing this will greatly improve your chances of getting the right speaker/amp setup. Here are some rules/guidelines to know:
A 3dB change in sound level is considered "barely perceptible" to the human ear. A 6dB change in sound level is a doubling (or halving) of the power (watts), but is still considered a small change in perceived sound level.
If you double the distance from the speaker to the listener then the sound level will drop to 1/4th, or -12 dB.
Typical speech over a PA system should be somewhere in the 65 to 85 dB range. A loud rock concert might be as high as 115 dB. Movies are in the 100-105 dB range.
Speakers have "sensitivity ratings". A typical rating would be something like 85 dB/Watt/Meter. Meaning that if you put 1 watt into it, and measure it at 1 meter, you'll get 85 dB.
So here's what this means... Let's say that your speaker has a sensitivity of 85 dB/watt/meter, and you are 2 meters away and feeding it 1 watt. The sound you hear will be 73 dB. If you go to 2 watts then you get 79 dB. 4 watts = 85 dB. 8 watts = 91 dB. 16 watts = 97 dB. 32 watts = 103 dB. 64 watts = 109 dB. 128 watts = 115 dB.
Now if you move the speaker to 4 meters away you drop down to 103 dB. To get back up to 115 dB you need a 4x in power, or 512 watts. The point is, very quickly you get into some serious power levels for just a modest increase in sound level.
All of this is irrelevant to the topic of needing a sub woofer. If you need more low frequencies, then get a sub. If you don't, then don't.
Parallelling identical 8 Ohm speakers to get a 4 Ohm load (and the opposite: putting 4 Ohms speakers in series to get an 8 Ohm load) is normal practice.
"The amplifier's power rating is greater than the sum of the speakers" - That can get problematic: at full power the speakers might be damaged. You want the speakers (combined) rated power to be at least the maximum power produced by the amplifier.
Best Answer
The largest cost in a DIY amplifier will be the enclosure if you want it to look halfway decent, then the transformer and heat sinks, etc. It also takes a long time to build. In your case I would advise a LM3886 chip amp, but read on...
Considering the speakers you are using, this isn't an audiophile project aiming for 0.00000 whatnot distortion, I'm sure you'd like something practical and decent sounding instead, with a minimum of fuss!
(Note: shopping questions are usually closed, so enjoy it while it lasts).
Get this, and a laptop power brick as a supply.
Rationale: the Chinese steamroller crushes everything in its path. Building your own is only worth it if you want world class audiophile hi-fi for ultra-megabuck speakers. This TI class D chip is known to work pretty well. And considering the price, honestly, why bother to drill a face plate and install connectors... drill & tap the heat sink... Gaaah! This'll work, probably a lot better than you'd expect from the price...
Don't make a fuss about the RMS power, if you put the speakers on the desktop near your PC monitor, you won't exceed a few watts anyway, that should be pretty loud already.
However:
Check the Philips amp that comes with the speakers doesn't use an active crossover! If this is the case then the speakers are married with it and you won't be able to use another amp. A simple way to check is to count the wires in the cable from amp to speakers: 2 wires is passive crossover (OK with any amp), if more wires suspect active crossover.