1. With the setup I would have with VOIP added to my network hardware (see above *running all 4 items at the same time*), would I need a Gigabit Ethernet capable router or not (the 10/100 Ethernet would be enough)? (What worries me is the *constant* Internet radio signal here. I do not want to experience any routine random transmission freezes.)
100Mbps should do. A standard VoIP call should not take any more than 128Kpbs. Your biggest concern should be your internet bandwidth. With 3Mbps to play with you may struggle with performance while streaming your internet radio.
2. The Wi-Fi Quality of Service software (QoS) standard feature help me here? If so, where would I put the emphasis on hardware making the most efficient use of Ethernet connections by prioritizing multimedia traffic (from highest priority to least priority)?
The QoS should help with the internet radio however the impact on other devices would be unknown. It may be worth considering implementing QoS for UDP traffic however most standard routers won't support that. You might want to prioritise your VoIP traffic as well so if you can try and get a router with this feature.
3. The speed for the Wi-Fi portion I find is rated theoretically at 300 Mpbs or a 150 Mpbs I typically find presently. Based on my hardware network setup *running all 4 items at the same time*, would I need to consider a Wi-Fi 300 Mpbs based router or just a Wi-Fi 150 Mpbs based (cheaper) router suffice? (*NOTE: Obviously, the older custom desktop computer will be hardwired to the prospective router and of course the VIOP landline box as well leaving my other two hardware network items needing wireless transmission.)
With only the devices that you have listed I think 150Mbps is suffice. Remember this is the local bandwidth! Every device that communicates on the internet will be forced through the 3Mpbs bottleneck at your ISP.
4. Is there any comments or suggestions you can recommend regarding a router I should consider? Please provide me those comments and suggestions.
With regards to security, A router with good port forwarding and access control is a given. Netgear have always had the nicest interface in terms of configuring security however that is just my opinion. Also, consider what the wireless range will be like and judge based on your physical building since you may have to get a repeater for full coverage.
I take it the VoIP box is independent and doesn't need a call manager? If not then you may need to look into more advanced enterprise routers however those are normally configured at a command line with no GUI as well as being costly.