Is $1+1 = 2$ true in any base?
Yes, this is valid in any base in which $1$ and $2$ are both digits (so, with the standard conventions, any base except base $2$). More generally, a single digit always represents the same number no matter what base you consider it in (as long as it is a valid digit in that base). So for instance, $3+4=7$ is valid when interpreted in any base (as long as the base is at least $8$, so these are all digits in the base).
To be more precise, we should be clear to distinguish numbers from the sequences of digits we might use to represent them. Standard notation unfortunately does not make this very clear. When we write $1+1=2$ normally, what we really mean is "the sum of the number represented by $1$ in decimal notation and the number represented by $1$ in decimal notation is the number represented by $2$ in decimal notation". So what "$1+1=2$ is valid in any base" really means is "for any base $b>2$, the sum of the number represented by $1$ in base $b$ notation and the number represented by $1$ in base $b$ notation is the number represented by $2$ in base $b$ notation." This is because, as mentioned above, "the number represented by $1$ in base $b$ notation" is the exact same number as "the number represented by $1$ in decimal notation", and similarly for $2$.
Integers have their own existence, separate from how we may choose to represent them.
The statement
"(the integer referred to by the decimal symbol 1) + (the integer referred to by the decimal symbol 1) = (the integer referred to by the decimal symbol 2)"
is indeed always true. Whether or not this is expressed in symbols as
"1 + 1 = 2"
depends on how you choose to represent integers.
That's a good question! According to the rules of arithmetic, 1 + 1 = 2 is a true statement about numbers.
Therefore, any reasonable way of representing numbers—whether base 2 or base 16 or base 10— should allow you to express that true statement.
On the other hand, maybe you're asking about symbols rather than numbers: maybe you're asking whether the symbolic expression 1+1 = 2 is true in any base $b$ (where we interpret 1 and 2 as symbols in base $b$).
The answer is yes: the statement is true in any base $b>2$. The reason is that for any base $b>2$, the symbol 2 is a meaningful symbol in base $b$; it refers to $2\cdot b^0$. And we have that $b^0 + b^0 = 2\cdot b^0$ by arithmetic— hence the symbolic expression 1 + 1 = 2 is true in any base $b>2$.
It is always valid because $1+1$ is the definition of $2$. It's not a theorem, it's what we mean by the symbol $2$ to begin with.
There are three forces at play here.
First comes our natural understanding of numbers. $2$ is a specific natural, or real, number. We understand it intuitively, and unambiguously. This is the abstraction of the idea of having two apples, two sons, two cats, or two examples for collections with two objects. As such, when we think of the number $2$, it is literally defined to be $1+1$.
Then comes the representation of a number in a particular base. This is now a question of how we dress the abstract number into a slightly more tangible form. Here $2$ is a relatively "bad" example, as most [natural-]bases are large enough so $2$ is just $2$ again. But think of the number ten. In decimal, this is $10$; in trenary this is $101$; in octal, $12$. All of these are different strings which represent the same number. How is this possible? Well, when we change base, we change how we interpret the string of symbols.
And finally comes the purely syntactical evaluation of symbols. This takes the last point and pushes it to $11$.(1) We forget that the symbols even represent the abstract quantity of how many hands a "standard" person should have. We only know that $1$, $2$ and $+$ are symbols in a mathematical language. As such we are in fact allowed to interpret them to mean pretty much anything that we want. You see this with $\pi$, which can denote a homomorphism, a projection, a function, a constant real number; but you see it less often with $2$ or $+$. Nevertheless, we are allowed to interpret those symbols to our liking, and we can easily concoct interpretations where $1+1$ and $2$ are not the same object.
In short, if you consider them as abstract numbers, they are independent of their representations, and then $1+1=2$, always. If you are asking about the interpretation of the symbols as numbers, then the answer is a qualified "usually". If you are asking in general about interpreting the symbols $1,2$ and $+$... then this is a resounding "not enough context".
(1) This can be taken in any base you prefer. Even in base $\omega$.