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Measuring one qubit in an entangled pair in another basis?

Yes, and no! If you have some basis change unitary $U$ for a single qubit, then the person holding the first qubit can perform that unitary on their qubit, using the mathematical description $U\otimes ...
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creating a generalised Bell state with gates

I think I nailed it. The 4 Bell states are $$\frac{1}{\sqrt{2}}\big [|00\rangle + |11\rangle\big],\quad \frac{1}{\sqrt{2}}\big[|01\rangle + |10\rangle\big],\quad \frac{1}{\sqrt{2}}\big[|00\rangle - |...
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creating a generalised Bell state with gates

Starting from any of the four basis states, Hadamard + controlled-not will produce each of the four Bell states. So, expressed in terms of the gates you need, you just need to add $X$ to either qubit ...
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