In the UK, it's pretty commonplace for Students leaving High School at 18 to go travelling on their own in Europe - especially as part of a 'gap' year before going on to University.
Having said which most 18 year olds will go with one or two friends. As the whole thing is more enjoyable if you share the experience.
But there are pros and cons to going with friends (I know of several groups that fell out of friendship and split up during the trip).
I would say that you definitely need to have a plan - research where you are going to stay and how much that will cost so you have enough funds. Also, the best value places to stay get booked up so you will want to have something set up in advance. Maybe not for every night of your trip, but at least for the first night in each city.
There are plenty of guidebooks and websites on this topic and you should spend time reading those - well before you go.
Like any part of the world, there are places you should avoid at certain times of day as they unnecessarily increase risk without adding enjoyment (again research online will tell you that), but from my knowledge of the UK at least, I would say you are pretty safe here.
Bear in mind that "Europe" is very diverse and each country has it's own distinct culture and norms - so you have to assess each country and indeed each city and locale carefully.
eg Marseille is different than Paris, central Paris is different than outer Paris and even within Paris certain areas are known as hotspots for Tourist muggings.
But all this sounds very negative and I don't mean to put you off as mostly you will have a lovely time and not experience anything unpleasant.
You will also likely come across other travellers (especially if you are staying at the Youth Hostels) and it's not uncommon to make lifelong friends on such trips.
You don't say if you are male or female - I think females on their own do attract unwanted attention sometimes and that can be a consideration - especially if you are not used to handling situations like that (Eg a bunch of drunk guys on a night out will often verbalise if they cross paths with a young woman - it maybe harmless and ignorable or it may escalate) . But sadly that is true of any country I have ever been in. On the other hand, statistics show that a teenage male is more likely to be the victim of a mugging in the UK say, than a woman.
As for your being an Eagle Scout that may give you a practical advantage, but they streetsmarts needed to stay safe can be a different skill set altogether.
Maybe a more relevant qualification is , have you travelled widely in your own country first? If not, do at least SOME travelling to another state or distant city in your own country first - you will learn valuable lessons of what it feels like to be alone and away from home - and you can judge better whether this is the right thing to do.
These days the ability to carry a mobile phone and stay in contact with loved ones and the possibility to have a credit card to avoid having to carry cash also reduces risks to you personally . And having a phone with GPS on it could be a godsend as (for me at least) one of the big questions was where exactly am I? If you have the internet on your phone then you also have a way of finding out things about local travel timetables etc and so yet another problem about travelling is resolved.
If you have gadgets on you though, do take care of them and don't flaunt expensive gadgets in certain areas where it will make you a theft target .
Knowing a foreign language is extremely useful as you will often get a lot more help from locals if you speak in their tongue. It's not that they don't understand English (often they do) but they really appreciate your making the effort to be part of their culture and so won't write you off as 'just another tourist'.
So that's a mixed bag of advice there - but in summary - yes it's doable and your budget is not unrealistic if you are frugal with your choices of where to stay and book well in advance.
I think you will enjoy it a lot more if you can go with some others though and it does add to the safety factor compared to being on your own.