Missing Stages
According to the Stage View, ordinary objects are temporally unextended timeslices. "Ted sleeps" is true iff the present Ted-stage sleeps.
What if there is no present stage, as with "Socrates is wise" and "Socrates exists"?
The question is not what to say about "Socrates was wise" and "Socrates did exist". These are true because at some time in the past, there is a wise Socrates stage (see pp.27f. of Ted Sider's Stage paper). The problem is the tenseless "Socrates is wise".
There are two possibilities. On the first, those sentences are false. I guess this would follow from the most straightforward Stage semantics of predication. But do we (eternalists) really want to say that, quantifying unrestrictedly, Socrates does not exist? And isn't it very odd to say that even though there are wise philosophers in 400 B.C. (for there are wise philosopher stages in 400 B.C. -- lots of them, actually), Socrates and Plato are not among them?
So one should better say that "Socrates exists" is true because "Socrates" is indeterminate between many person stages, all of which exist. On this account, "Socrates is wise" and "Socrates sleeps" may either be true or false or truth-value-less, depending on what we say about cases where only some resolutions of the indeterminacy satisfy the predicate.
If "Socrates and Plato are among the wise philosophers of 400 B.C." is to come out true, one will have to say that a "Socrates"-sentence is true iff it is true on at least one resolution of the indeterminacy, which is rather unusual. It will also make both "Socrates sleeps" and "Socrates does not sleep" true.
It may also seem slightly counter-intuitive that when I say "Socrates is a Greek philosopher", I'm not talking determinately about one particular person. On the current proposal, I haven't made up my mind which of infinitely many distinct persons I mean. An adequate response to "Socrates is a Greek philosopher" would be, "it depends on who you mean by 'Socrates'" (even if it's clear that I was not talking about the prime minister of Portugal).
At any rate, if "Socrates" is indeterminate between all Socrates-stages, then "Ted" should also be indeterminate between all Ted-stages. (Otherwise the semantic value of a name would drastically change the moment its bearer dies.) This is interesting because it contrasts with modal counterpart theory. Counterpart Theory doesn't say that "Ted" is indeterminate between the actual Ted and all his counterparts.
Still, what past objects are for the Stage View, fictional objects are for Counterpart Theory. But the difficulties are less severe in the latter case. For instance, if only some of the Sherlock Holmes candidates satsify some predicate P, saying that it is indeterminate whether Holmes is P is intuitively correct (as opposed to saying that it is indeterminate whether Socrates is one of the great philosophers of 400 B.C.).
Is there a problem with the following naive idea: read (tenseless) "Socrates is wise" as "Socrates is, or was, or will be wise". And whatever story you tell about the truth of the past-tense "Socrates was wise" now secures the truth of the tenseless statement, through securing the truth of one disjunct.
I'd like to hear a bit more about those past tense statements involving things that no longer exist, though. I thought that "N was F" was analyzed through the F-ness or otherwise of a past counterpart of the *present* N. (Just as in the modal version truth conditions depend on counterparts of *actual* individuals). But that won't fit a case where there is *no* present N.
(By the way, the (copy of the) Sider paper to which you link doesn't have a p.27!)