Dear Aleksee, i'm sorry that I have to tell you this but it is near impossible to force a higher version of android onto your phone.
Rooting wouldn't really solve anything, it would only make your phone less secure. Running Cyanogenmod with android 4.0 or higher (android 3.x is only for tablets) would make your phone really slow and would reduce the battery life (in addition to any other bugs that may occur).
Android is an amazing Operating System; you can get it on phones, tablets, media players, netbooks, E-readers, etc. An advantage of Android is that it's Open Source (google "GNU license" for more info) and free, so its easy for manufacturers to use it.
A disadvantage of this is that Android needs to be modified for use on certain devices, for copyright and advertising reasons.
Basically, when Google releases a new version of Android, Google Nexus devices (like Sandal100's) will receive it immediately.
What companies like LG, Samsung, HTC, Motorola (owned by google, but still), Huawei, etc. have to do is change it to support the hardware in their devices (some components are open source, some are closed source, and the minimum hardware requirements of the new version of android, like the minimum amount of ram must be met) and manufacturers also change it to make it "unique", by changing interfaces, blocking off certain settings and installing their own apps (Samsung for example has ChatOn, kies, group play and allshare).
Long story short: you have two options;
1. Admit defeat and see if you can use some other device (like a computer) to run the app or try to find a similar app. Android 2.3.6 is still a good OS, and lots of people still use it because lots of app developers still support it (Android SDK lets you choose what versions you want to support).
2. Buy a new device (not recommended unless you plan on buying a new device anyway). My advice would be a Nexus (LG Nexus 5 phone, Asus Nexus 7 or Samsung Nexus 10) as you get the best value for money and get the longest software support.