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Easy differences arise if one allows principal ultrafilters, since the ultrapower of $X$ by a principal filter is canonically isomorphic to $X$, but other ultrapowers are not. Another easy difference arises when $I$ is uncountable, since one filter might concentrate on a countable subset of $I$ and others might not, and this can dramatically affect the size of the reduced power, making them different.

So the question is more interesting when one considers only non-principal filters and also only uniform filters, meaning that every small subset of $I$ is measure $0$.

In this case, under the Generalized Continuum Hypothesis, the ultrapower of any first order structure is saturated, and thus any two of them will be canonically isomorphic by a back-and-forth argument. Without the GCH, it is consistent with ZFC to have ultrafilters on the same set leading to nonisomorphic ultrapowers.

Also relevant is the Keisler-Shelah theorem, which asserts that two first order structures---such as two graphs, groups or rings---are elementarily equivalent (have all the same first order truths) if and only if they have an isomorphic ultrapowers.

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For question 1, the anwer is yes

Easy differences arise if one allows principal ultrafilters, even since the cardinality ultrapower of the reduced power can depend on the $X$ by a principal filter is canonically isomorphic to $X$, but other ultrapowers are not. For exampleAnother easy difference arises when $I$ is uncountable, since one filter may might concentrate on a countable subset of $I$ and another may others might not, and this could dramtically can dramatically affect the size of the reduced power. Another example would arise from the fact that the reduced powerby a principal ultrafilter essentially gives you $X$ back, but other reduced powers do not.

For question 2, the situation is more interesting. Again, one filter could concentrate on a much smaller subset of $I$ than another, and again this could make the ultrapowers making them differentsizes. Also again one should consider only nonprincipal ultrafilters to avoid the other counterexample from before. Thus,

So the question is more interesting when one considers only non-principal filters and also only uniform ultrafilters on $I$, filters, meaning that all every small subset of $I$ is measure 1 sets have the same cardinality as $I$. 0$.

In this case, under the Generalized Continuum HypotheesisHypothesis, all the ultrapowers will be isomorphic, even when $X$ is a ultrapower of any first order structure . The reason is that the ultrapower will be saturated, and so thus any two of them will be canonically isomorphic by a back-and-forth argumentwill provide the isomorphism. In general, without Without the GCH, however, it is consistent with ZFC to have ultrafilters on the same set leading to nonisomorphic ultrapowerscan differ.
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For question 1, the anwer is yes, even the cardinality of the reduced power can depend on the filter. For example, one filter may concentrate on a countable subset of $I$ and another may not, and this could dramtically affect the size of the reduced power. Another example would arise from the fact that the reduced power by a principal ultrafilter essentially gives you $X$ back, but other reduced powers do not.

For question 2, the situation is more interesting. Again, one filter could concentrate on a much smaller subset of $I$ than another, and again this could make the ultrapowers different sizes. Also again one should consider only nonprincipal ultrafilters to avoid the other counterexample from before. Thus, the question is more interesting when one considers only uniform ultrafilters on $I$, meaning that all measure 1 sets have the same cardinality as $I$. In this case, under the Generalized Continuum Hypotheesis, all the ultrapowers will be isomorphic, even when $X$ is a first order structure. The reason is that the ultrapower will be saturated, and so a back-and-forth argument will provide the isomorphism. In general, without GCH, however, the ultrapowers can differ.