Before we continue talking about hardware, I think it's important that you have a think about the way a proper multiroom music system works. It'll help you understand the limits of what you can do on a tight budget, and it'll help you plan a path so that you don't waste time and money buying things that are going to be no help further down the road.
Any real multiroom system can do the following basic functions:
* Allow 'per room' volume control
* Provide a way to group rooms together to form zones e.g. master bed + master bathroom become a zone
* Allow 'per zone' volume control (with the ability to still adjust the rooms to get the right sound balance across a zone)
* Provide a way to group all rooms/zones together as 'whole house'
* Allow 'whole house' volume control with the ability to trim the volume in each room for sound balance
* Make it possible to play any of the music sources in any individual room or grouped zone and control that independently of the rest of rooms/zones
* Play one source throughout the entire house (Party Mode)
If you went out tomorrow and bought a true 12 channel multiroom power amp then you still wouldn't be able to all the zone selection/grouping and volume control, and that's because a lot of the functionality comes from the source devices. Your system would be based on a single source player. In essence then, whatever you do based on a single source player, you're always going to be working in Party Mode until you add more players. That's fine so long as you accept that playing internet radio or Spotify in the kitchen means it's also going to be playing in the lounge and bedroom and bathroom at the same time.
The next thing to consider is the size of any amp and the heat it'll generate. There's some electronics/physics to factor in here.
The main consideration is power loss due to the thickness (or thinness, if you prefer) of the speaker cables. Put simply, fat, pure-copper speaker cables pass more of the Wattage generated at the amp end. Couple them with high efficiency speakers (88-90dB/W/m) to make the most of the available power.
Speaker cables tend to come in several nominal sizes. For home cinema and multiroom audio applications, 4mm is about as thick as is generally used. Then the sizes range down in steps from 2.5mm, 1.5mm, 1.0mm and 0.75mm. The amount of power loss is proportional to cable length, and inversely proportional to thickness. If we say that you have an amp capable of generating 30W per channel, and you put that down 50m of copper cable, then with the 4mm stuff you'd only lose 10% of the power. Drop down to a 2.5mm cable and the losses increase to a shade over 15%. At 1.5mm the losses jump up to 25%. That means our 30W of power ends up at roughly 22W when it reaches the speaker.
These are figures for a solid copper cable, but not every DIY or budget install uses solid copper. Some people buy copper coated aluminium cables (CCA) because they're much cheaper. Some are aware they're buying something inferior. Other times they're not. Aluminium is a much poorer electrical conductor. The thin coating of copper anodised on the surface does nothing to change this. Some web sellers try to play the pseudo-science card and talk about skin effect. It doesn't have any benefit until the signal frequency goes way above the audio range. The bottom line is that CCA cables lose an additional 40% of the power compared to solid copper. Our 22W going through a 1.5mm copper cable becomes just 13W when travelling via CCA. That means well over half the power is lost.
This is why you can't get away with low-powered cheap T amps to drive multiroom speakers. They don't have enough power to start with to cope with the cable losses, especially if CCA cable is involved.
Next, speaker efficiency. Pyle doesn't quote a dB/W/m figure. That means one thing; the speakers aren't that efficient. Again then, you'll need plenty of power in the amp to cope with the speaker's lack of efficiency.
What you're looking for then is a AV receiver or AV amp with a 5.1ch/6.1ch/7.1ch input and plenty of balls. Some suggestions include the Denon 38xx and higher series, Pioneer AV amps (VSX-AX5i, VSX-AX10, SC-LX series), bigger Yamaha AV receivers RX-V7xx series and above, and the RX-A-series AV amps (some of the older 4xx and 5xx series might also fit the bill as they have strong power supplies and MCH inputs), Marantz, some Sonys e.g. STR-DA1200. I would avoid Onkyos as they have a bit of a reputation for being fragile. Ditto ARCAM plus they go for a small premium die to the Hi-Fi roots, and that's not really needed here. Try to keep the amp purchase under £100 if possible.
You don't need HDMI, but I wouldn't rule out anything with HDMI inputs as long as the price is right. There's a Denon 3808 on Ebay right now which would tick a few boxes. A MCH output isn't required. All you really need is MCH IN, plenty of raw power (TIP: Look at the user manual. Take the mains power consumption figure. Knock off 10% and then divide by the number of speaker channels to get a rough idea of the real power output. e.g. the 3808 eats around 860W according to the Yank specs. 860 x 0.9 = 774W It's a 7.1 channel receiver, so divide by 7. This gives around 110W/ch)
How you'll hook up and run is like this. Buy a Monocor stereo to mono line level converter plus whatever jack-to-phono and phono-to-phono cables you need. Connect the Dot to the Monocor, then the Monocor to one of the stero inputs on the receiver. (You'll use MCH once you add more Dots). Hook up a small TV to the monitor out on the amp/receiver. You'll need this for the On Screen Display to do the set-up. Reset all the speaker channels to 0dB or do a system reset. If required, do an input allocation. Connect the speakers. Set the amp/receiver to All Channel Stereo. (Once you've finished doing the settings, and you're happy with the levels of two kitchen speakers and the bathroom vs bedroom, then the TV can be disconnected.)
What you have now is the Dot playing everywhere, in mono. Set the general speaker level with the AV amp/receiver, and then use the Dot's volume control for day-to-day adjustment.
When you're ready, buy two more Dots and two more Monacors. This is where you'll swap over to the MCH input. The Dot for the kitchen goes to say the front L&R connections (make sure the speaker outputs follow suit.) Dot for the lounge goes to center. Dot for the bedroom & bathroom goes to Surround L & Surround right. Change the amp to the MCH input. Balance the speakers again if required. The MCH input turns the AV amp/receiver in to a sort of multichannel power amp. Now each Dot can run independently. This means you have 3 zones: Kitchen (2x speakers in mono - or you could go stereo there without the Monocor in-line), Lounge (1x speaker mono), Master bed & bathroom (2x speakers in mono).
That's you up and running. The rest of running the system is down to how well the Dots run as multiroom music sources.