Go to beddy-bye' means 'go to bed' in the sense of 'go to sleep,' but 'beddy-bye' alone does not mean a bed in the sense of a piece of furniture, and there is no such word as 'beddy.' An infant sleeps in a bassinet or carrier, a toddler in a crib or a baby bed. ('Cot' is BE only in this sense.) A bigger bed can be called a 'big-girl bed' or 'big-boy bed' (even if it's still a little smaller than full-size).
The ending 'a-bye' also appears in very old baby language having to do with sleep. Possibly the best-known lullaby (lull- + -a-bye) in English is 'Rockabye, baby, in the treetop.' There are some lullabies in which 'bye-bye' also means sleep, and you could probably say 'go bye-bye' for 'go sleep (but not *'go to bye-bye').
But in modern English 'bye-bye' is much more commonly a short form of 'goodbye,' and baby talk includes things like 'go bye-bye' (= leave), 'say bye-bye' (= say goodbye), and 'wave bye-bye' (= wave goodbye).
Repeating words is sometimes a feature; for example, 'say night-night' (= say goodnight) works the same way as 'say bye-bye.' And Archfarchnad is right about forming rhymes ending in -y. Things like 'itsy bitsy,' 'teeny weeny,' and 'Humpty Dumpty' show the pattern, and you can make up others to be funny, though 'forky-worky' might be pushing it a bit far.
'(Not by the hair on my) chinny-chin-chin,' to the best of my knowledge, is a quotation. Parents should be able to name the story. (-; Seriously, one of the best ways to expose a child to a foreign language is through nursery songs, rhymes, and stories.
Animals are more likely to have diminutives if they have names and known sexes and are called 'he' or 'she' accordingly. So 'puppy,' 'doggie,' 'kitty,' 'horsey,' etc. are fine; similarly, children themselves have childish nicknames (which they will then have to outgrow as adults), like 'Tommy' or 'Ruthie.'
But as Archfarchnad says, you don't have much emotional connection to a functional object like a spoon or a bed, which is always just 'it.' And it's obviously little, so it would be redundant to say 'your little spoon' or 'your little bed.' Unless you're just seeing the nursery furniture for the first time and oohing and ahing over it ('Ohhh, look at that sweet little wocking chaiw, isn't that just the most pwecious thing!') But don't go overboard; Archfarchnad's health warning here, needless to say, is against nausea. (-;
I've never heard of a 'botty' either, and I agree that '-kin(s)' is either archaic or BE. And I support the usual advice: Avoid creating your own cutesy endings, use 'little' only in extreme moderation, and in most cases, just use the normal words. Children start recognizing words earlier than you might think, and they might as well learn the right ones. You will be close enough to baby talk if you just repeat things a lot with emphasis on the key words (Where's your spoon? Where's that spoon? Do you see a spoon? Who can see Caitlin's spoon?), and exaggerate emotions by changes in facial expression (big eyes, high eyebrows) and in the pitch of your speech. Hard to describe online, but if you have a baby, you're probably already doing it. (-: