Prof.Richard Feinberg writes:

Existence.
I don't know of any precise equivalent.  Usually this is indicated by a tense marker.  Or one might say eiei (e=present, i='in', ei='it) or e tuku, which means something like 'to be preserved' or 'to be intact'.  One might also use the word tu 'to stand' or nopo 'sit' or 'stay'. 

Eiei ni vaka i te penua nei? 'Are there any canoes on this island'? 
Te vai e tu?  'Is there a spring?' 
Ko te pare koi tuku, pe ku marepe?  'Is the house still there, or has it been destroyed?'

Existing….I'd give same ans. as for 'existence'

Life……Mauri or ora.  Koi mauri pe ku mate?  'Is it still alive, or has it died?' Ora is more like 'spirit' or 'soul', or sometimes 'lifestyle'.

Being [noun]………A being is te mea, literally, 'the thing'.  For being in the sense of being vs. nothingness, Anutans tend to use the verbal form.

Being [verb]……..see 'existence'

I was………Ne kau....  (ne = past; kau = 'I')  Or i te vaatia ne kau...  'when i...'
E.g., I te vaatia ne kau nopo i Kalakana....  'When I was living on Guadalcanal....'

He was……..Ko ia ne... or i te vaatia ko ia ne....  E.g., Ko ia ne kau i te kanotau 'He served (or He was serving) in the army.

They were……Naatou ne....

I am……..Kau... or E kau....  Kau maanatu; koee, kau nopo 'I think; therefore, I am' or more literally, 'I think; thus, I am present'.

Same pattern for items below; the critical indicator is the personal pronoun plus (sometimes) a tense marker.  For present tense, the verbal particle is often omitted.

You are.....................
He is.........................
They are.....................
I will be......................
He will be...................
They will be................
There was..................
There will be...............
It existed....................

Finally ten short sentences. [Please put the 'existence' word in brackets.]

Example: The girl [is] a student.

The girl is a student….Ko te tamatii papine e ako 'The female child is learning'.

The boy is being kind…….Te tamatii tangata e aropa.  'The male child is acting with compassion'.

The building has existed for ten years……Ko te pare ne tu nga poi tau e puangapuru.  'The building stood for ten years'.

The girl is running quickly…..Ko te tamatii papine e rere ke vave.  (or e puri aru 'fleeing' or rakaraka 'striding quickly')

The boy will be handsome…….Te tamatii tangata ka taurekareka.

The man was a doctor………Ko te tangata ne dokta.

I believe in the existence of atoms……….No indigenous word for 'atom'. 
Possibly: Kau maanatu ko nga atom e tuku (or e turaki) 'I think atoms are around' (or 'are standing').

Life exists on earth…….Nga mea e mauri e pare i atangata nei. 'Living things are present in this world'.

I like being a student………Kau vakivaki i te vaatia kau ako i te pare skulu. 
'I'm happy when I'm learning at the school house'.

I enjoy living my life…..I imagine an Anutan would say something like: Kau vakivaki i atangata nei.  'I'm happy in this world (as opposed to the world of spirits)'.  I'm not aware of any equivalent to 'my life', and they'd probably view 'living my life' as redundant.

I love him/her with my whole 'Being.'….Toku aropa ki ei kairo oti.  'My affection for him/her has no end'.

 
It would be better for you to ask a native speaker, but I think my answers are pretty much correct.  In any case it has been an interesting exercise.

Good luck.

Rick Feinberg.

Richard Feinberg.
Professor of Anthropology.
Kent State University.