Sincerely, I could find no trace of a magazine Sharps rifle; it was possibly a prototype specifically made for those trials that never passed into production once it was rejected...
As for why lever-action repeating rifles (holding more than 1 round) were never adopted by any military, I couldn't tell why... my guesses can be:
#1: Being them originally conceived for uses other than military, for saddle carry, to be fired while riding a horse, etc., their dimensions were contained by using cartridges that were simply not powerful enough to guarantee range, accuracy and stopping power required by the Military. Only exception was the Winchester Model 1895
, which by the way was significantly longer and heavier than other lever-action models and de facto nulled all the major advantages of the lever-action carbines.
#2: As can be deduced from above, a lever-action rifle chambered for a military caliber looses most of its advantages as a practical, light, handy rifle, as its working system is most of times mated to a tubular magazine, which makes the gun significantly long and bulky if it has to fit Military cartridges. Again, the Winchester Model 1895
is an exception, using a standard internal magazine patterned after the one of most of the bolt-action military rifles of the era (compared to whom, it was also significantly faster to operate). But again, due to the resistance and robustness required to handle such heavy cartridges, it offered almost no advantage in lenght and weight, nor in ammunition capacity (the Winchester Model 1895 holds only 5 rounds of ammunition, is over 1 Meter long and weights around 4 Kilograms UNLOADED). Furthermore, the Winchester Modek 1895 rifle was more sensitive to fouling and dirt than rifles like the Mosin-Nagant M-91 or the Mauser 98G, and the lever action was less comfortable to operate when firing from prone position, so typical for early 20th Century warfare, and it was also long to reload: while the Russian contract Model 1895s used Mosin-Nagant 5-rds stripper clips, the other variants were reloaded by single cartridges, round by round.