He is so cute!
First off, if a vet thinks neutering will change aggressive behavior you should look for a new vet. That is such a discredited myth. If she doesn't know that what else doesn't she know?
Sounds like he is a bit of a butt head. When we took our five puppies in for their first shots, the four girls were like champs. Then we had "Jungle King". OMG. Took three vet techs and the vet to give him a shot. When we left people in the waiting room wanted to know what had happened he was so loud. Totally ridiculous. Not as ridiculous as your vet's suggesting your pup be neutered, but close!
It was the same for toe nail trimming. What a wuss.
On to the fear reaction: We kept one of the girls. Very gentle and mellow. We decided as part of socialization to leave her at doggie day care for one day a week. The first day we dropped her off we took her to the enclosure and left her. When we came to pick her up they told us to wait in the parking area below because people coming back agitated the dogs. We heard shrieking from hell, looked at each other, and said: "That sounds like it might be Diki." After more shrieking the main woman came out carrying Diki and told us she had biten one of the handlers when she went to pick her up. Apparently she thought she was going to be carried to her death.
Long story short this was the only time she has been aggressive. A good lesson that even a sweet natured Basenji will bite if they feel threatened enough. Not to say that all dogs would have interpreted the situation as she did. Just to say that strange environments can be very scary to a puppy. IOW I wouldn't be too concerned about this at this age.
Nails can be an issue. Try a grinder (like a Dremel). Not sure it matters for the dog but at least you don't take too much off and have them bleed. FWIW we've always given baths when they start to smell funky. They hate it but it's the price they have to pay to sleep in the bedroom with someone who has an exceptionally good sense of smell (not me).