You are right. Let me edit my post.

Dear Alex, your question wasn't more elementary, because you asked for some interpretation of the Tamagawa number conjecture in terms of natural objects. To find this might be a difficult task (especially since natural can be in the eye of the beholder). However, specializing Tamagawa Number Conjectures to the case of modular forms is an interesting exercise, not at all easy by the way, but certainly doable (and done 15 years ago). One last thing to Ian: the references I mentioned should also do for twists by characters but you should know that for some twists, things get much harder.

Dear Ian, In the case of a normalized eigenform over $\mathbb Q$ (which I gather is your case of interest), then everything can be done explicitly, and in fact has been. Two good references are K.Kato Iwasawa theory and p-adic Hodge theory (Kodaira math. 1993) and $p$-adic Hodge theory and values of zeta functions of modular forms (Asterisque 2004). That said, there is another place where everything is checked quite explicitly, and this is the PhD. thesis of M.Gealy. The first and third references are online (I think) ut if you have trouble finding them, you could always e-mail me.

"how one goes about picking the right sublattice of the units?" My very limited understanding is that properly formulated, the beauty of these conjectures is that they don't depend on the choice of lattice. Have you read the articles of D.Burns and D.Burns and W.Bley on the subject?

Merci beaucoup.

The answer to question 1 is yes: you want to look at the works of Tilouine, Genestier-Tilouine and most recently V.Pilloni. The article of J.Tilouine which appeared in Compositio Math 142 would seem to be a good starting point for you. Note that typically, these results show p-adic modularity, in the sense that they do not establish that the form is really classical, as you wish. The so-called classicity property is much harder, but might be within reach.

The strategy of Blasius-Rogawski to construct a motive for Hilbert modular forms (base change to unitary then transfer to U(3) then to an inner form) is indeed not known to succeed in this case (because of the shape of the L-packets). I don't know if this strategy and its close cousin are known to fail (way back, Blasius and Rogawski were actually "cautiously optimistic" that it should succeed by transfer to U(4)). But you surely knew this, as you also surely know that the meromorphic continuation of the L-function of E as in your introduction is known anyway.

"Tout ce dont tu rêves y est démontré." Quelle belle phrase...

All of them are potentially modular (this is because there exists a rational prime l such that E has good ordinary reduction at every prime in F above l). To prove that a specific example is actually modular might be hard, as all the works I am familiar with assume either residual modularity or large image (but I am definitely no expert either).

I think it more polite to let experts and the usual course of events decide whether specific parts of an unpublished work is correct or not. I think that MO is a terrible place to do this, in particular because anonymous and pseudonymous comments are possible.

No one seems to have mentioned explicitly the Quillen-Lichtenbaum conjecture, which gives a full answer. You should look it up, especially as it is now probably a theorem after the work of Voedvodsky and Rost.