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 $b>2$.