Looking at these as alternatives without synonymous meanings, I've marked which are ok or not, and made a few other little corrections for you:

a)  Tiffany hardly ever travels abroad - ok.( Tiffany fairly travel abroad - not ok).

b)  The package was rather small ( the package was too small / the package was quite small). - all ok

c) This task is very easy to do ( this task is quite easy to do ). -  both ok

d) That man drives too fast for his age ( that man drives rather fast for his age ). - both ok

e) Poor families have barely enough to eat ( poor families have hardly anything to eat). Ok with the shown corrections, but not using barely/hardly on their own.

Yes it's true that fairly can mean moderately - but we wouldn't say that someone travels moderately abroad either.

Don't forget that these words are not really moderating the 'travel' but the 'frequency' and you have neglected to use any frequency words. That's why your sentences were incorrect. 'Tiffany travels abroad fairly often' would be ok (but moderately doesn't fit in this context). That's why I added 'ever' after hardly.

I don't think we can come up with a 'complete' list of these words for you but here are a few more for you:

slightly - completely - totally - occasionally

Learning which one(s) are appropriate in any given context is the hard part.

Anonymous wrote:

Hi!  Yes, Yoong Liat ! Thank you very much for your reply. By the way taking a look at my list of adverbs of degree is there anything missing ? Could you please, help me to let my list more completed ?  Very  - very much - too - a lot - extremely - rather - a little - a bit - rather - fairly - quite - barely / barely enough - hardly - can hardly - hardly ever - hardly any - quite a lot - absolutely - completely - scarcely - ..........................................................         Juliette

seldom, often