In the Busoni edition, they seem not to indicate this at all (notice the dolce and legatissimo marks)
There are two ways of thinking about this kind of edition. First, you can view it as a "derivative work." That is, in your role as a recreative artist, you can set yourself the task of realizing the editor's interpretation as a work in its own right. You are presenting to your audience not only Bach's composition but Busoni's interpretation. This isn't particularly common; indeed I don't remember ever encountering it, but if I saw such a recital advertised I'd probably buy a ticket. I'd certainly click the link to the video.
Second, you can see it as suggestions. Since tastes have changed since Busoni's time, you can take the suggestions or reject them. Don't ask yourself only whether you agree with each suggestion; try to go behind them to the reason. For example, perhaps Busoni intended a crescendo to highlight a given formal inflection point in a fugue; perhaps you agree that the point requires highlighting but prefer to do it with a diminuendo. Or maybe it looks as though Busoni added a slur somewhere to highlight a certain phrase structure and you disagree with his analysis; in that case, maybe you still want a slur but you will place yours differently.
I do not think I have the right to play it like this, i.e. long durations more legato than the shorter ones.
As others have noted, you have a right to do whatever you want. It is not a matter of dogma. However ...
as I am a conservatory student, I do not think they would let me play like this.
That is a different matter. Your teacher and your examiners may indeed see it as a question of dogma. Of course, one person's dogma is another's heresy. They might condemn your interpretation while a teacher at the next conservatory might praise it. Whatever you do, you should not ignore the possible consequences of going against their teaching. Some teachers would welcome such independence; others would not.
If you conclude that defying your teacher might be risky, perhaps you could develop two interpretations of the piece: spend a few weeks playing it as you'd like to, then switch to your teacher's preferred approach. You'll surely learn something more this way.
my question is: is it totally "illegal" to perform this piece as I have indicated? or could there be good reasons for it?
There certainly could be good reasons for it. There are multiple baroque-era sources saying that the normal duration of a note in the absence of any other considerations should be half of its notated value. Some modern Baroque specialists underestimate (in my opinion) the frequency of other considerations and as a consequence their performances sound (again in my opinion) somewhat trite. It's a fundamentally unvocal approach, and another thing you often find in period sources is the ideal of emulating the human voice. So playing everything staccato obviously can't be the whole story.
I did a very unscientific survey of recordings available on YouTube, and to my surprise I couldn't find even one aside from Gulda's with staccato sixteenth notes. I was hoping to find one because I don't particularly like Gulda's interpretation, and I'm convinced that there is an interpretation with staccato sixteenths that I would like if only I could find it. My problems with Gulda's performance is partly that I find it too slow and partly that it seems a little artificial, insufficiently organic, and overly metronomic, and I disagree with the execution of the trills and other ornaments (trills should generally begin on the upper note). But everyone else sounds like they're playing a nocturne from the 19th century -- even many of the harpsichordists.
I echo 'yo's suggestion to move beyond the words "staccato" and "legato": think of the 16th notes as articulated and the 8th notes as singing. If you're leaving a bit of space between the sixteenth notes, play with the amount of space you leave. The notated articulation in your example hints at this, with the first two slurred and the rest staccato. But the staccato notes shouldn't all be the same length. The odd-numbered ones should be a bit longer. Maybe they're not so long as to warrant a slur on the page, but they can still be a bit longer than the even-numbered ones.
The same holds true of the eighth notes, if you decide that you need to play them staccato for your teacher's sake. For me at least, "staccato" implies a sharpness of attack that you could perhaps avoid. Think "detached" instead, and above all recognize that the duration of the detachment is variable, and you can use that variation to shape the phrase. On historical principles, those on beats 1 and 4 should normally be longer than the others, and I think you'll find that observing this fits with the construction of the counterpoint and the way dissonance and consonance align with the meter.
Finally, consider the instrument you're playing. The degree of articulation you use with the sixteenth notes or eighth notes could be different on a concert grand or a baby grand, on a tall upright or a short one, since the effect of the damper dropping on the string will be different. Similarly, you will want to adapt your articulation depending on the size of the hall you're playing in and its acoustics. Something that sounds choppy in your practice room might be best for a large palace hall with marble floors; something that sounds muddy in a small empty recital hall might work just fine once it is full of people.