The Rules
It is always good to go back to first principles and look at what the rules actually say. From the Basic Rules (pp. 33-34) (Player's Handbook is identical I believe; p. 122), I have highlighted what I consider to be key points:
Alignment
A typical creature in the worlds of Dungeons & Dragons has an alignment, which broadly describes its moral and personal attitudes. Alignment is a combination of two factors: one identifies morality (good, evil, or neutral), and the other describes attitudes toward society and order (lawful, chaotic, or neutral).
[...] Individuals might vary significantly from that typical behavior, and few people are perfectly and consistently faithful to the precepts of their alignment.
Alignment in the Multiverse
For many thinking creatures, alignment is a moral choice. Humans, dwarves, elves, and other humanoid races can choose whether to follow the paths of good or evil, law or chaos.
Alignment is an essential part of the nature of celestials and fiends. A devil does not choose to be lawful evil, and it doesn't tend toward lawful evil, but rather it is lawful evil in its essence. If it somehow ceased to be lawful evil, it would cease to be a devil.
The overarching concept is one of choice (unless you are a devil etc.); to my mind, the alignment written on a character sheet is aspirational - this is the type of person the character wants to be; not who they are.
This contrasts sharply with the earliest editions of D&D where the consequences of consistently acting outside alignment were severe (loss of a level in AD&D - a fate worse than death for a character!)
The problem of Evil (and Good)
How do you recognize an evil (or good, lawful or chaotic) act? Are they objectively quantifiable or are they contextual?
The problem is akin to that addressed by Justice Stewart when considering pornography:
I shall not today attempt further to define the kinds of material I understand to be embraced within that shorthand description [hard-core pornography]; and perhaps I could never succeed in intelligibly doing so. But I know it when I see it, ...
Similarly, evil (or good, lawful or chaotic) is hard to define but I know it when I see it!
Part of the problem is that even after several thousand years of thinking about this stuff, there is no consensus of what "good" is: is it altruism, utilitarianism, liberalism, egalitarianism, something else? Similarly with "law", do we need: democracy, communism, feudalism, socialism, something else?
I contend that the ethical content of an act is a combination of its consequences and the intent of the perpetrator. This is reflected in most judicial systems: to be a crime the perpetrator must have criminal intent.
Under the RAW, an act done by an angel is both Lawful and Good simply because an angel is doing it. Equally, the same act is both Chaotic and Evil if done by a demon for the same reason. It is within the DMs purview to decide that there may be acts that one can perform which the other is incapable of but I argue that these should be the exception rather than the rule. For example, both should be able to comfort or kill a child in the right circumstances; in either case an angel acting for the greater good and a demon acting on a titillating whim.
When the act is done by a PC, things really get confusing.
Reading the description of the alignments as they apply to those with moral agency, it is clear that you do not have to be vicious to be evil nor does being good mean you can't be vicious. As written, good is synonymous with altruism, evil with selfishness, law with community and chaos with individuality. An evil character may kill with regret, a good character with satisfaction, what matters is the motive for the killing.
The Specifics
The druid is acting bang on to his personality, he has no reverence for society, either the broader state or his current companions: chaotic to the core. Although he is clearly putting self interest first here, it is clear that he does not consider depriving his companions of their "fair share" to be either evil or good: in his mind it doesn't hurt them and he can use the money for useful stuff like saving the forest. The question of goodness comes when the fighter or paladin needs the money for something - will the druid give it to him?
You give no information about the the paladin except his alignment. He is clearly not acting to his alignment; he is not showing loyalty to the fighter (chaotic) and is acting in the paladin's own self interest (evil). Oh well, the paladin set high ideals for himself and failed to live up to them, don't we all? Maybe he will feel guilty about his small acts of Chaotic Evil, maybe he won't. Of course, maybe he doesn't like the fighter, maybe he feels he has earned the extra money, maybe he feels the fighter would just blow it on drink while he can use it to help the poor (that is, the poor who are not innkeepers).
Unless he is breaking his oath (which he isn't because he hasn't taken it yet) there are no in game consequences. You have some interesting opportunities, however, when he does reach 3rd level: perhaps he can't take his oath. Some quests to find out why and to make amends might be in order, hmmm?
Summary
To my mind, alignment for PCs is a role-playing guide like traits and flaws. I consider them as aspirations of who the character would like to be and I don't see it as the DMs role to castigate them when they turn out to be someone else so long as everyone is having fun.
If you really want to meddle in this, then some positive reinforcement through inspiration when the character plays to their alignment at a cost is the way to go.
Best Answer
In 5th edition, alignment has no mechanical effect whatsoever. It is useful only as a roleplaying aid.
No, but if his alignment is dramatically different from that of their god, then it becomes an interesting topic to explore in-game. Why do they worship that god? What do they do if a priest of that god asks them to do something that their alignment would oppose? etc.
As mentioned, there are no mechanical benefits.
Non-mechanical benefits might include:
And to take the word "character" in another sense:
As with any background detail, the act of selecting a god to worship (or rejecting them all) prompts a player to think about who the character it. This helps give them a character (as opposed to a sheet of numbers).