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Ulysses is written in iambic pentameter. There are a few spondees and trochees thrown in for good measure, but I'm confused in some places, like here:

I cannot rest from travel: I will drink

Life to the lees: All times I have enjoy'd

Greatly, have suffer'd greatly, both with those

That loved me, and alone, on shore, and when

Thro' scudding drifts the rainy Hyades

Vext the dim sea: I am become a name

In the second line, there's a trochee (Life to), but then it continues on normally. Ok. Fine. The line after that does the exact same thing (Greatly). What goes on in the fourth and sixth (last) lines, though?

That loved me, and alone, on shore, and when

The first pair (that loved) is an iambic foot, the second (me, and) is what? A trochee? The third (alone) is, finally, an iamb. The rest are iambs.

What goes on in Tennyson's head when he creates such a weird structure? How does he decide to break the rules, and where? Also, is it correct to think in pairs when breaking down a poem like this?

Similarly, the last line is also funky:

Vext the dim sea: I am become a name

Vext the: trochee.

Dim sea: spondee?

There's also a line at the end of the poem that has 11 syllables, but I guess he just thought "shit, this sounds so good I just have to leave it in" because it's the best line in the entire thing. But I'm guessing that's not the usual thought process going on?

Tennyson was indeed writing iambic pentameter.

Certain substitutions are traditionally allowed in iambic pentameter, namely, a foot can be replaced by a trochee or a spondee, and two adjacent feet can be replaced by a double iamb.

Here are some examples.

Trochaic substitutions: much have and cities are trochee in the lines:

Much have I seen and known; cities of men

And manners, climates, councils, governments,

Trochaic substitutions are especially common in the first foot of a line, but are allowed in all the feet except the last one.

Spondaic substitutions: day wanes, moon climbs, moans round are all spondees in the lines:

The long day wanes: the slow moon climbs: the deep

Moans round with many voices.

A double iambic substitution, where two adjacent feet have been replaced by a pyrrhus and a spondee (two unstressed syllables followed by two stressed one); for example, which in óld dáys in the line:

We are not now that strength which in old days

Finally, I assume your 11-syllable line is

Moved earth and heaven, that which we are, we are;

This probably wasn't originally supposed to be an 11-syllable line.
When Tennyson was writing, it seems to have permissible to treat the word heaven as one syllable (but not similar words like leaven). This reflected a peculiarity of pronunciation that has disappeared since. See this English stackexchange question.

Although most iambic pentameter uses the above substitutions, some poems adhere to a particularly strict version of iambic pentameter. For example, in Edna St. Vincent Millay's sonnet I will put chaos into fourteen lines, the only deviation from perfect iambic pentameter are trochaic and spondaic substitutions in the first foot of a line.