Satan

There is no theological reason why it is impossible to see Shaytan, although most people do not see Shaytan on a regular basis. 

However, how does your friend know it is Shaytan, specifically? Is he arriving with an ID card? What about just garden-variety shayateen?

Basically there is no way to prove things which are not objectively measurable. That is, there is no way to prove he is seeing Shaytan and there is no way to prove that he is not. All you can say is that he sees something, and this is what he understands it to be. 

It is said that doing wrong acts leaves an ugly imprint on the soul which becomes manifest in the hereafter. For this reason, some mystics are said to be able to see people's true natures. So perhaps this might also be what he is seeing. 

I might be suspicious of the clause "whenever someone does something wrong according to Islam", because sometimes people do things that are ethically wrong but are not obviously forbidden. For instance, it is allowed to buy a knife. However, if I buy a knife to murder someone, this is very wrong. Still, it would not be obvious from my purchase that I am committing a sin.

Similarly, we do not always know when someone is doing something wrong. For instance, someone might be lying or trying to flatter someone in order to take advantage of them, but we do not always know it. Along the same lines, we don't usually know if someone is secretly breaking their fast or committing other hidden sins.

(Of course if he is seeing Shaytan in these cases, it is a different story.)

Anyway, most people do not require Shaytan to do wrong things as we are quite capable of doing them ourselves without him. 

There is nothing in Islam that connects 666 with the devil. The symbolism between 666 and the devil developed in history among Christians over time. There is a reasonably comprehensive discussion of this on Wikipedia: https://en.wikipedia.org/wiki/666_(number) 

Still, it is unusual if you are seeing that everywhere. You should be sure it is not just observer bias. For instance, a person might see many numbers in one day but pay more attention to that number because it is meaningful and a little scary. If it is really showing up everywhere (for instance, several times a day in random places) and an objective observer would agree with it, that is definitely weird, although it's hard to say what that might mean. 

Anyway, it seems like the underlying root of your question is whether you are just having bad luck of an ordinary kind, or whether there is something metaphysically unhealthy going on (for instance, the Qur'an discusses the evil eye).

Bad luck can indeed just sometimes be ordinary and due to decisions that don't work out (such as an investment or job that doesn't work out), the everyday travails of life (needing expensive dental work), or circumstances out of our control (such as how the coronavirus has affected the global economy).

(This is apart from genuinely bad choices which lead to bad results, but I'll assume that's not your case here.)

Sometimes what appears as bad luck is also a way that God guides us to do something else through fate. For instance, I try as hard as I can, but I can't get a job in a certain city or field, so I get a job in a different city or field, and this is par of my fate and a way of moving me on to something else. So, is it possible you are being guided towards something else? 

And sometimes there are spiritual/metaphysical components, such as the evil eye, although that is less common. Anyway, if that is something you are concerned about, you could discuss it with a religious scholar who has knowledge of these things and who is trustworthy and see if they have any thoughts on your situation. 

Also, don't forget, if you aren't doing so already, there are du'as and surahs you can recite to seek an increase in sustenance from Allah, for instance, Surah al-Waq'iah on Thursday nights. You can read more on duas.org.

Hope things get better for you!

A lot of things are said about what is and is not possible regarding dreams about Ahl al-Bayt (A), however, as far as I know, most of them have no basis.

It is possible for the Shaytan to appear as something he is not; in fact, the main feature of Shaytan is deception. 

However, even if you were dreaming about something other than Ahl al-Bayt, there are a lot of things between Ahl al-Bayt and Shaytan that you could have been dreaming about. (Including your own subconscious!)

If, in your dream, the being told you to do something evil or which you know is wrong, it is good to ignore it. It doesn't really  matter where it came from. It could have been Shaytan or something else. 

If it told you to do something good, it is good to accept it.  

If it told you something else, you can also look into it and see if it is a beneficial or destructive thing. 

If it just stood there and didn't do anything, then it doesn't really matter - all you can say is, you don't know. 

So, I don't think you should worry too much about Shaytan being in your dreams. While it is good to be conscious of Shaytan's existence, it is not good to get too paranoid about it (indeed, that could also be a trick of Shaytan!). I think Shaytan is probably busy in other places and maybe doesn't have too much time for dreams unless there is some very specific reason why you think this is the case. 

(This is assuming you actually had a dream about this and are not just asking in general)

It is said that the garden that Adam and Eve were in was a different garden than the one that people to go eternally in the afterlife.

Allameh Tabataba'i brings up this question in Tafsir al-Mizan (available online) in the section on narrations 2:35-39 and discusses it in light of the narrations. For more elaboration on this question, as well as relevant narrations, where the garden might have been, and so forth, you can read that section.