Do classes have the same memory placement and memory alignment format
  as structs?
The memory placement/alignment of objects is not contingent on whether its type was declared as a class or a struct. The only difference between a class and a struct in C++ is that a class have private members by default while a struct have public members by default.
More specifically, is data loaded into memory based on the order in
  which it's declared?
I'm not sure what you mean by "loaded into memory". Within an object however, the compiler is not allowed to rearrange variables. For example:
class Foo {
    int a;
    int b;
    int c;
};
The variables c must be located after b and b must be located after a within a Foo object. They are also constructed (initialized) in the order shown in the class declaration when a Foo is created, and destructed in the reverse order when a Foo is destroyed.
It's actually more complicated than this due to inheritance and access modifiers, but that is the basic idea.
Do functions affect memory alignment and data position or are they
  allocated to another location?
Functions are not data, so alignment isn't a concern for them. In some executable file formats and/or architectures, function binary code does in fact occupy a separate area from data variables, but the C++ language is agnostic to that fact. 
Generally speaking, I keep all of my memory alignment and position
  dependent stuff like file headers and algorithmic data within a
  struct. I'm just curious to know whether or not this is intrinsic to
  classes as it is to structs and whether or not it will translate well
  into classes if I chose to use that approach.
Memory alignment is something that's almost automatically taken care of for you by the compiler. It's more of an implementation detail than anything else. I say "almost automatically" since there are situations where it may matter (serialization, ABIs, etc) but within an application it shouldn't be a concern.
With respect with reading files (since you mention file headers), it sounds like you're reading files directly into the memory occupied by a struct. I can't recommend that approach since issues with padding and alignment may make your code work on one platform and not another. Instead you should read the raw bytes a couple at a time from the file and assign them into the structs with simple assignment.