Learn English – “cut down on” vs “cut down”

In this context, I would say “please have a seat”. You're inviting the person to sit down, but not formally request that they sit down, merely offering the possibility.

“Please take a seat” is also possible, but slightly less deferential. It tends to be used in contexts where the person really should sit, for example by a doctor talking to a patient.

“Please sit down” and “please be seated” are more directive. For example, an air hostess might tell a passenger on a plane to “please be seated”, because the passenger really must seat while the aircraft is in motion. However, tone is key: it is possible to say “Please, sit down” (or perhaps that should be “Please! Sit down!”), with a pause after “please” and the right intonation, to convey that you are happy to see the person and cordially inviting them to sit down.

To cut, as you already know, is generally 'to slice with a sharp object'; if you "cut your own arm", you take a knife and make an incision somewhere on your arm.

To cut out is probably, in this case, 'to free from entrapment by cutting the trapping material'. If you "cut out your own arm", then somehow one arm is trapped, perhaps under a fallen log, and you use your other arm to cut the log until you can get your first arm free. (This usage seems very unlikely.)

The most common use of cut out is 'to shape an item by cutting a larger element'; for instance, if I have a piece of paper, I could 'cut out a paper doll'.

To cut off is to sever or detach; if you "cut off your own arm", then either you no longer have an arm, or you need major surgery to get it re-attached.