Does "classification" of mathematical objects have a precise meaning?

I don't think it has a precise meaning, but usually to classify a set $X$ means to express $X$ as the disjoint union of the images of a some functions $$f_0 : P_0 \rightarrow X, \;\ldots, f_n : P_n \rightarrow X$$ For example, for the classification of closed surfaces, we let $X$ denote the groupoid of all connected closed surface, and we define:

1. $f_0 : 1 \rightarrow X$ returns the sphere
2. $f_1 : \{1,2,3,\ldots\} \rightarrow X$ maps $g$ to a connected sum of $g$ tori
3. $f_2 : \{1,2,3,\ldots\} \rightarrow X$ maps $k$ to a connected sum of $k$ real projective planes

One can then show that for each $x \in X$, there exists unique $i \in I$ such that for some $p \in \mathrm{dom}(f_i)$, we have $f_i(p) \cong x$. This fact, which amounts to saying that the images of our functions partition $X$, is called the classification theorem of closed surfaces.

This is different to being able to tell whether any two objects specified in any given language are equal/equivalent/isomorphic/whatever. Such problems are usually called "isomorphism problems" or "recognition problems" - consider, for example, the graph isomorphism problem from graph theory, or the recognition problem from knot theory.

So the answer to your questions are as follows:

1. Does "complete classification" of some objects necessarily mean that we can write an algorithm that determines whether two input objects are isomorphic, where the algorithm completes after a finite number of steps?

No. For example, there's a good classification of closed surfaces, but if you specify a surface in a sufficiently rich language, you're not going to be able to work out what standard-form object it's isomorphic to.

2. Suppose I have a finite list of properties such that iff A and B agree on all properties, then A and B are isomorphic. But suppose the "value" of one such property takes on a type of object that is not completely classified. [...] Does this mean I would still not have completely classified manifolds?

Reducing one isomorphism problem to another isomorphism problem does not necessarily solve the former unless the latter has already been solved. It's still important progress though.

3. Suppose I don't have a list of properties that uniquely determine each object (up to isomorphism), but instead have a list of properties that still somehow provide a nice organization scheme. Is this ever considered a classification?

As long as each class in our scheme can be realized as the image of a function that is considered "constructive" and "well-understood", it would be considered a classification.

I'll also remark that a lot of classification theorems aren't recognized as such even though, strictly speaking, they fit the above pattern. For example, the set $X = \{(x,y) : x^2 + y^2 = 1\}$ is classified by the function $f_0 : [0,2\pi) \rightarrow X$ given by $f_0(\theta) = (\cos \theta, \sin \theta)$. Yet few would call this a classification of the points on the unit circle. You sometimes here the word "parametrization" thrown around when only one $f_i$ is involved in the classification.