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Jibran85!
1...
Some OEMs (like Google itself with its Nexus phones) limit the sharing of apk files via BlueTooth (or even NFC). The easiest solution to get around this, is to rename the .apk file as .mp3 or .jpeg ... and once its transferred, you can easily rename it back. The other option would be to use a 3rd Party tool.
2...
Do you get any error or anything?
3...
Common problem with such handsets (Vega or LG LTE Phones). Its mostly because of the Charger. I'd recommend getting the Samsung Original Charger.
Works well with most of the devices.