"Visne" not "Velisne" would be more common. The principal parts are Volo, Velle, Volui, but the conjugation is slightly irregular.
For some reason they appear to be using the subjective, velis. I don't recall this as a common use of the subjunctive in Latin.
Using 'velim' is more polite. In French you would say 'je voudrais' instead of 'je veux'. In German 'ich möchte gern' instead of 'ich will'. You can say 'volo' in Latin, but that is very direct and impolite. The coniunctivus is a polite way of saying that you want something.
I understand other European languages have this politeness factor, but Latin (classical) does not. All of these sentences should be using indicative forms of volo, not the subjunctive.
In the first place I am very impressed with your posts. However I am curious, since no one speaks Latin and all we have to go on are the texts, how do we even know about the subjunctive?
I am not positive how it is used, but one way to tell would be from ancient written dialogue. There are no native speakers of classical Latin, but there were plays written in Latin.
Makes perfect sense to me. So if the subjunctive is used in Plays then it would seem to me that using it here would be acceptable.
The recording is very difficult to understand due to fast pacing of words and enunciation.