First off, welcome to Globe Community. To answer your question:
1. Based on the model of your Samsung Device this model comes from Cricket Telecom, the reason you are not able to get LTE it's because your device does not support Globe's LTE frequency. Upon checking your phone only supports LTE frequency bands: LTE B2/B4/B5/B12/B25 while Globe LTE uses B3. However, you should be able to at least get 3G or HSDPA speed. Check your APN settings, if you are using Globe Prepaid sim the correct APN should be http.globe.com.ph if you are on a Postpaid the correct APN should be internet.globe.com.ph
2. As for purchasing Apps from Google Play I know there are still issues with this, I suggest that you try calling the hotline using your Globe mobile dial 211 or using landline dial 730-1000 and report your concern.