JGCON, for whatever it is worth here is my experience with these type of devices. I am not disputing what chattphotos said above, no doubt that is what he has encountered. My experience however is a little different. I live in a fairly large L shaped house. I do not get much interference from neighbors since I have some land around me. While I can see some of their signals, they are weak and usually not an issue. I mostly have to be concerned with own devices, cordless phones, security cameras, etc which are all broadcasting on the 2.4 band. I have a Cisco dual band router in my office which is towards one end of the house and I have a second access point in the master bedroom to serve that end of the house. The router is quite powerful and reaches both floors and even the garage at 2 to 3 bars (mostly 3). In the bedroom I was getting 1 bar (sometimes dropped) and in the bathroom further down I usually got dropped on all my iOS devices. I then purchased a Cisco Wireless Power Line Adapter to serve as a secondary AP (TP-Link sells the same) and I can now serve that end of the house at 2 to 3 bars. For far so good....
Initially I set my Router and AP to different SSIDs and channels thinking first, that I could tell which of the two radios I was connected to by looking at the SSID on the IOS device, and secondly, that in that fashion I would have less interference or conflicts between the two. With that configuration, if my iPhone was locked to the router at 3 bars, as I walked to the far end the signal would degrade and I would only connect to the AP once I was dropped from the main router. There would be periods of weak signal and momentary lags while switching networks. However, after I configured both the Router and AP to the same SSID and channel I now get a seamless transition, i.e., if my phone is locked with the router at 3 bars and I walk towards the far end, as the router signal degrades to 2- and the AP is now at 2+ or 3 the phone will make an absolutely seamless transition between the two, no signal ever lower than 2 or any drops or lags. I cannot tell with of the two radios I am connected to but it is irrelevant (actually I can - by now I know that when on the bedroom wing I am always on the AP).
As to why your extender does not allow you to receive emails while connected to it (it should of course), I would try Forgetting both Networks or better yet, Reset Network Settings and reconnecting to both to see if that makes a difference. The other alternative is to delete you Outlook account, re-start your phone, and then add it back. The fact that you can get your emails while on the router makes this reset less likely to work. What happens when you walk to within 6 feet of the router? Does the phone stay locked to the Extender? Also knowing about your configuration (SSID, channel, band, etc.) would help.
If interested here are links to the Cisco Wireless Adapter I am using and the TP-Link equivalent unit. I know you already bought an extender but the TP-Link PL Adapter (~$50) would be another option. See TP-Link's description of the differences below. TP advises better performance with a PL Adapter than with an extender (but possibly some issues with circuit breakers, see the first review on the Cisco).
http://www.amazon.com/Linksys-Powerline-Wireless-Extender-PLWK400/dp/B0087O6AQE/ ref=sr_1_4?ie=UTF8&qid=1440777577&sr=8-4&keywords=wireless+powerline+adapter
http://www.amazon.com/TP-LINK-TL-WPA4220KIT-ADVANCED-Powerline-Extender/dp/B00HS QAIQU/ref=sr_1_1?ie=UTF8&qid=1440778556&sr=8-1&keywords=tp-link+wireless+powerli ne
