I have an IceCrypt STC3250 set top box, Not what I would call a good box, it does have draw backs. However it claims to have three inputs, although I only use Freeview and FreetoAir will not call it Freesat as the EPG is rubbish. It will allow one to record to a hard drive in a format which can be read with other devices without having to do a special format on the hard drive.
Mine around 4 years old got it from Maplin. It does some times fail to record, and it does not use standard channel numbers, however since the Freeview and FreetoAir are all listed together it could not really follow standard numbers. If Freeview is scanned first then that does follow standard numbers.
Since I don't have cable in this area not tried it with cable. The network cable was plugged in and it did at one point show internet weather, but that now seems to have stopped.
I have the same with Sony Blu-ray it did work, but now very limited internet on it. However lucky for me the two latest TV's will show U-tube etc.
As to recording it seems some makes limit what can be recorded, we have two old LG TV's with built in hard drives, they will record Freeview, but not Sky. Had the same with a DVD recorder, first one would record anything, then the replacement would not record anything it considered copyright protected, in some cases even my own home movies it saw as copyright.
It has been pointed out how services listed on the box of TV's and set top boxes may be correct at time of manufacture but now may have changed.