Anyone who tells you that a speaker can be damaged by under-powering it clearly knows nothing about speakers or electronics. Worst case is that you don't get the sound quality or volume you were hoping for.
Yes, an active sub-woofer has its own amplifier. This is what you would typically use for a decent home theatre setup. Passive sub-woofers would only be used in the cheapest home theatre setups or in some car sound systems.
You've used the word amplifier through your post but I believe you are looking at an AV receiver. This can decode digital audio formats and amplify for speaker level outputs but it will also pass through video to your TV. I'll use the word receiver below, but you can think of this as an amplifier if you like.
Pioneer make some equipment that is ok, and some that is cheap rubbish.
Is this the receiver you are looking at?
http/www.pioneerelectronics.com/PUSA/Home/AV-Receivers/Pioneer+Receivers/VSX-520-K
300 euro is too much to pay for this receiver. Recommended retail price is only USD $229.
Output is 80W per channel, which is ok.
This receiver has no speaker calibration. Better receivers allow you to place a microphone in the normal sitting position and the amplifier will automatically adjust volume levels of the speakers to compensate for differences in distance and output.
The surround speaker connections are the spring clips I was complaining about earlier too, not a deal breaker but not a good sign.
If looking at Pioneer, there is a newer model VSX-524-K for RRP USD $249.
This has speaker calibration, with the same power output as the VSX-520-K.
It also has 4 HDMI 2.0 inputs rather than 3 HDMI 1.4a inputs on the VSK-520-K.
http/www.pioneerelectronics.com/PUSA/Home/AV-Receivers/Pioneer+Receivers/VSX-524-K
The VSX-824-K is RRP USD $399. This adds screw terminals for surround speakers and a bunch of connectivity features that may or may not be useful.
http/www.pioneerelectronics.com/PUSA/Home/AV-Receivers/Pioneer+Receivers/VSX-824-K
The brands mentioned by rexter are all good; Marrantz, NAD, Harman Kardon and Sherwood. I also suggested Yamaha. Generally the base models are a bit limited, especially from Yamaha or Sherwood, so often next model up is better.