Learn Hardcore Finnish: Minä haluan levätä ainakin tunnin.

Minä haluan levätä ainakin tunnin. – fully correct, a bit more emphatic on I.
Haluan levätä ainakin tunnin. – also fully correct and very natural.

You typically include minä when:

you want to emphasize the subject:
- Minä haluan levätä, enkä sinä. – I want to rest, not you.
you contrast people:
- Minä haluan levätä, mutta hän haluaa työskennellä.

So: yes, you can (and often will) drop Minä in this sentence.

What exactly is the form haluan, and how is it related to the dictionary form haluta?

Finnish dictionaries list verbs in the basic form (1st infinitive), here haluta (to want).

Haluan is:

present tense
1st person singular (I)
indicative mood (neutral statement)

Conjugation of haluta in the present indicative:

minä haluan – I want
sinä haluat – you (sg.) want
hän haluaa – he/she wants
me haluamme – we want
te haluatte – you (pl.) want
he haluavat – they want

So the pattern is: verb stem halua- + personal ending (-n, -t, -a(a), etc.).

What form is levätä here, and why isn’t it something like lepään?

Levätä is also in the basic form (1st infinitive) = to rest.

The structure is:
(Minä) haluan + [verb in basic form]

So:

haluan levätä = I want to rest
haluan syödä = I want to eat
haluan nukkua = I want to sleep

If you conjugate levätä for I in the present, you get lepään (I rest), but you do not use that form after haluan:

Minä lepään tunnin. – I rest for an hour.
Minä haluan levätä tunnin. – I want to rest for an hour.

So after haluan, use levätä, not lepään.

What’s the difference between levätä and nukkua?

levätä = to rest, relax, recover
- may include lying down, sitting quietly, not working
- you might or might not sleep
nukkua = to sleep
- explicitly means being asleep

Examples:

Haluan levätä ainakin tunnin. – I want to rest (not necessarily sleep).
Haluan nukkua ainakin tunnin. – I want to sleep at least an hour.

So if you specifically mean sleep, use nukkua; levätä is broader.

What does ainakin mean exactly, and how is it different from vähintään or ainakaan?

Ainakin means at least in the sense of a minimum, maybe more:

Haluan levätä ainakin tunnin. – I want to rest at least an hour (maybe more).

Differences:

ainakin – neutral at least
- Join ainakin kolme lasia vettä. – I drank at least three glasses of water.
vähintään – more formal / numeric, no less than
- Tarvitsemme vähintään kymmenen ihmistä. – We need at least ten people.
  In speech you can often use ainakin instead.
ainakaan – used in negative sentences, meaning at least / anyway in that negative sense
- En ainakaan halua levätä. – I, at least, don’t want to rest.
- En halua levätä tuntia ainakaan. – I don’t want to rest for an hour, at least.

So in your positive sentence, ainakin is the normal choice.

Why is it tunnin and not tunti or tuntia?

Tunnin is the genitive/accusative singular form of tunti (hour).

In Finnish, when you express a specific duration of time with many verbs, you typically use this genitive-like form:

Lepäsin tunnin. – I rested for an hour.
Olin siellä viikon. – I was there for a week.
Odottelin sinua puoli tuntia. – I waited for you for half an hour.

Compare the forms of tunti:

nominative: tunti (dictionary form)
genitive: tunnin
partitive: tuntia

In this sentence:

tunnin → for an hour (a whole, concrete block of time)

Tunti (nominative) would normally be used when tunti is a subject or a predicate noun, not a duration object:

Tunti on pitkä. – An hour is long.

Tuntia (partitive singular) can appear, but with different structures or nuances. For a simple for an hour with levätä, tunnin is the natural, standard choice.

So:

Haluan levätä ainakin tunnin. – correct and natural.
Haluan levätä ainakin tunti. – incorrect.
Haluan levätä ainakin tuntia. – sounds wrong/unnatural in this meaning.

Can I move ainakin somewhere else, like Haluan ainakin levätä tunnin? Does the meaning change?

Yes, you can move ainakin, and the focus changes slightly.

Haluan levätä ainakin tunnin.
– Neutral, default word order.
– Focus on the duration: at least an hour.
Haluan ainakin levätä tunnin.
– Now ainakin leans more on haluan / levätä.
– Roughly: At the very least, I want to rest for an hour (even if nothing else happens / I get nothing else).
– Slight emphasis on resting as your minimum requirement.

In most everyday contexts, version 1 (as in your example) is the most natural. Version 2 is possible, but more about contrast or emphasis in a larger context.

How would I say “We want to rest at least an hour” instead?

Change only the form of haluta to 1st person plural:

Me haluamme levätä ainakin tunnin.
– We want to rest at least an hour.

You can also drop Me:

Haluamme levätä ainakin tunnin.

Pattern:

Minä haluan… – I want…
Me haluamme… – We want…

How do I say “I want to rest for at least two hours” in Finnish?

You change the number and adjust the noun form:

Haluan levätä ainakin kaksi tuntia.

Notice the changes:

kaksi (two)
tuntia – after numbers 2 and up, Finnish normally uses partitive singular of the noun.

So:

1 hour: tunti → tunnin in the duration expression
- Haluan levätä tunnin. – I want to rest for an hour.
2 hours: kaksi tuntia
- Haluan levätä ainakin kaksi tuntia. – I want to rest for at least two hours.

What tense is haluan, and how would the sentence look in the past?

Haluan is:

present tense,
indicative mood,
1st person singular.

To make it past tense, you conjugate haluta in the past (imperfect):

Minä halusin levätä ainakin tunnin.
– I wanted to rest for at least an hour.

You could also drop Minä:

Halusin levätä ainakin tunnin.

How do I make this sentence more polite, like “I would like to rest for at least an hour”?

You can use the conditional form of haluta, which often corresponds to English would like:

Haluaisin levätä ainakin tunnin.
– I would like to rest for at least an hour.

Here:

haluaisin = conditional, 1st person singular (I would want / I would like).

It’s softer and more polite than Haluan levätä ainakin tunnin, which can sound quite direct.