Do you have some exact measurements of your computer desk? Size of monitor? Other stuff that will take up space on the desk?
Reason this would be good for us to know is that you really want the biggest main driver/speaker you can fit on your desk. I wouldn't go any smaller than a 5" main driver if you want decent low-end output for big Contra Bass sound in classical music. I DO have back-ported active monitors, and, yes, you have to have some distance between back of speaker and wall. The distance will depend on your room though. Mine sound best at about 12" from the back wall. Isolation pads/stands are recommended as well to keep the desk from vibrating. These will also tighten up the sound as well.
I think an X-ART ribbon tweeter is what you'll want for classical. The one in the Emotiva Airmotiv compares to the one in Adam active speakers. I used my Airmotiv 5 with just the analog out on my computer for a while. You'll be fine with compressed music codecs but you'll want a DAC for high-resolution music. There is A LOT of classical out there in SACD or 24-bit/192 kHz quality.
As others have stated, the Emotiva Airmotiv 6 are currently on sale because these are being phased out in favor of their Stealth 6. They are a bit too big for my 60" x 30" desk with 27" monitor and computer, but they are solid performers with 200w RMS per speaker. I can't recommend the Emotiva DC-1 DAC enough. It is a well-designed DAC with precision analog volume control and TWO high-quality headphone pre-amps. It also has a standard RCA analog input for legacy sources. Other DACs to consider are, of course, Topping, Dayton, Teac, Audioengine, and Parasound.
I haven't heard the Adam F7, but I did a side-by-side with the A5X vs. my Emotiva Airmotiv 5. I just couldn't justify the price of the Adam when I really couldn't tell the difference those and the Emotiva. I preferred the Emotiva over the rest of the stuff I demoed at Guitar Center as well.
Denon AVR X7200WA, Emotiva XPR-5, Emotiva XPA-7, Sony XBR 75" 850C, Oppo BDP-103D, Roku 4.
DefTech BP7000SC (L/R), CLR3000 (C), SM55 (FH), UIW BPZ/A (SB), UIW 75 (TM).
JBL L7 (Sur), EoSone RSR 350 (RH).
PC 2.1 setup. Emotiva Stealth 6, SVS SB2000 sub, Emotiva DC-1 DAC.