The short answer is that charging with a granny charger (3 pin plug) is fine provided your electrical infrastructure is in good shape. Manufacturers would not produce systems to do this unless it was safe.
The caveat in the above statement of course is "provided your electrical infrastructure is in good shape". A granny charger will draw about 10A. There is nothing pluggable in your house that will draw that amount of current for that long a time (many hours to charge). Kettles and toasters maybe, but for short periods. Tumble driers possibly the only thing that will come close, cookers will, but are normally hardwired. In other words, nothing is going to test out better whether or not your wiring and connections are in good shape.
And let's face it, a lot of houses have electrical wiring that is in a pretty shocking state and has not been rewired for half a century and checked in 20 years.
I have recently acquired a hybrid and use the granny charger off a dedicated spur. You need to be careful if for example you are running it off a ring, because if you are running other high power devices off the same ring that may cause the fuse to trip, or worse overload the wiring without the fuse tripping if it is in a bad state. When I ran mine for the first time I ran it on half power (5A) and checked everything was not getting too hot. I then started running at 10A. The sockets and wires (in what I consider to be a well maintained installation) rose in temperature by about 5C, so you can tell that the system is being worked.
What I would not do is plug in a system on full power for the first time on an old and poorly maintained electrical installation and then skip off merrily down the park for 8 hours with your kids and then leave it to the gods to decide whether by the time you come back your house isn't a smoldering wreck. If you have any doubts at all about the integrity of your wiring it should be checked out by a competent electrician beforehand in my opinion.
Re the chargers, you can get I think 3.2kW and 7.2kW installed at home easily. 3.2 will charge about 1.5x faster than the plug (2.4kW), 7.2 3x faster. So faster but still not particularly quickly. When the dedicated charges are installed the electrician will make sure the wiring is highly rated to account for the demand. I think the DNO (which people incorrectly call the "National Grid" on here) need to be notified for a 7.2kW charger. Not so sure about the 3.2kW. You may need the DNO fuse (next to the meter) upgraded from 60A to 100A for the 7.2kW and if your house is on a shared supply that may need to be split. I do not know whether the DNO charges for this or not. Personally I think getting a wall box installed is a great way of making sure your infrastructure is up to the challenge and that the wiring is all correctly rated. There are also some other things about wallboxes that make them more suitable for long term charging, but you can look then up on the web if you are interested.
Re the range, I can charge about 30 miles in 4 hours. So with a granny charger you should easily be able to charge 300 miles per week.
Re the cost, obviously you will save more with a tariff, but you do not need one to save money.
My last car was a diesel and very efficient at 50 mpg. That works out to about 12 miles per litre. One litre of diesel is about £1.50, so that is about 12.5p a mile.
At the current price cap electricity is about 30p per kWh. My car takes 4 hours @ 2.4 kW to charge for 30 miles, ie about 10 kWh per 30 miles, or 3 miles per kWh which works out to be about 10p per mile. So cheaper than the very efficient diesel and hopefully the electric price will come down soon.