Think of your phone as a person. Every time you run another app, you're giving that person another pound to carry. Depending on the ability of that person, they will be capable to carry many apps at a time, but if you don't use your task manager to close any apps you're not using, then you just are adding unnecessary weight to the amount the person has to carry - which will slow them down and make it hard for that person to do normal day-to-day tasks. Keep up with your task manager, and you shouldn't have any problems.
I would think no internet connection would be related to your phone company not the phone.