What's the tense for repetitive past action?

You don't need to replace it with anything as it is officially a perfectly grammatical usage of would:

Would (3): Used to; was or were habitually accustomed to ( + bare infinitive); indicating an action in the past that happened repeatedly or commonly. [from 9th c.]

The example from the Wiktionary:

When we were kids we would sit by the radio with a tape recorder on a Sunday, listening out for the chart songs we wanted to have. ~ 2009, "Soundtrack of my life", The Guardian, 15 Mar 09

Although as you can see, the dictionary states used to as an alternative, therefore if you feel the need to, you may rewrite your original sentence to:

On Thursdays, we used to go to the dump to shoot rats.

And still retain the original meaning.

The usage of would you describe is simply the past tense of will. It is perfectly legitimate.

Would is a modal auxiliary verb expressing, primarily, unreal meaning (as in your first example), and, secondarily, habit (as in your second example), prediction and volition. As a modal verb, it is invariable and cannot helpfully be said to be in any tense.

Your sentence If I were sleepy, I would go to bed is an example of what is known as the Second Conditional. (The First Conditional would be If I am sleepy, I go to bed and the Third Conditional If I had been sleepy, I would have gone to bed.)

Yes, would can be used for the imperfect tense in English (also known as the "past continuous" or "past progressive" tense.)

The following constructions could also be used:

On Thursdays, we used to go to the dump to shoot rats.

On Thursdays, we were going to the dump to shoot rats.

English only has two real morphological tenses left, Present and Past. And they're not really all that well distinguished; English morphology is almost all gone now.

All verbs, except modal auxiliary verbs, have a Present tense form (is, am, are, has, have, goes, go, set, etc.) and a (usually different) Past tense form (was, were, had, went, came, set, etc.). The present tense form is equivalent to the infinitive form, except that the 3sg form (subject he/she/it) is inflected with the {-Z₃} 3spres suffix. Those other things that people call "tenses" aren't, I'm afraid. Further discussion of English tenses and non-tenses can be found here.

However, would is a modal auxiliary verb, and all English modals are defective, lacking infinitive forms (I want to can do that), past participles (I have must do that), and past tense forms for the most part. The old past tense forms have become modals in their own right, with their own syntax, idioms, exceptions, and curiosities, though they can still function as past markers, in some constructions.

More importantly, all English modals are ambiguous; each has at least two kinds of meaning:

• Deontic, involving human social phenomena like obligation, permission, ability, desire
• Epistemic, involving logical calculations about possibility and necessity

More details on modals can be found here.

Would was formed from the past tense of a pre-defective will, and has many of the same base senses, somewhat modified. In particular, it can still function as the past tense of a deontic habitual use of will, which means, after all, "want, desire, be willing"; it's the epistemic use of will that means "future prediction", and this is normally repetitive in sense, because that's characteristic of things one does willingly:

• The old man will spend hours just staring at the ocean. (at some current time(s)
• The old man would spend hours just staring at the ocean. (at some previous time(s)

That's the deontic sense of would; it involves desire and willingness.

The epistemic sense is the conditional one (the subjunctive mood verb form, when English had a subjunctive mood, was based on the past tense form of the verb, so this is a normal development). But this isn't a conditional.