There is already an excellent answer to this question by Edlothiad (I don't think I have ever managed to answer a Tolkien question before him). As the question lists language as one of the things that Hobbits and Dwarves have in common, I want to add a little about that.
The language that Hobbits and Dwarves have in common is not their own but rather Westron, the language that the Dúnedain brought from Númenor, which was known as the Common Speech as it was known to all races.
The Hobbits adopted Westron when they migrated north to Bree and the Shire.
It was in these early days, doubtless, that the Hobbits learned their letters and began to write after the manner of the Dúnedain, who had in their turn long before learned the art from the Elves. And in those days also they forgot whatever languages they had used before, and spoke ever after the Common Speech, the Westron as it was named, that was current through all the lands of the kings from Arnor to Gondor, and about all the coasts of the Sea from Belfalas to Lune. Yet they kept a few words of their own, as well as their own names of months and days, and a great store of personal names out of the past.
The Lord of the Rings Prologue, Section 1: Concerning Hobbits
Page 4 (Single volume 50th Anniversary Edition)
Earlier, when they lived in the Vales of Anduin, they spoke the language of the Men of those part. That explains why Théoden recognises the word hobbit when he first meets Merry and Pippin.
The Riders laughed. ‘It cannot be doubted that we witness the meeting of dear friends,’ said Théoden. ‘So these are the lost ones of your company, Gandalf? The days are fated to be filled with marvels. Already I have seen many since I left my house; and now here before my eyes stand yet another of the folk of legend. Are not these the Halflings, that some among us call the Holbytlan?’
‘Hobbits, if you please, lord,’ said Pippin.
‘Hobbits?’ said Théoden. ‘Your tongue is strangely changed; but the name sounds not unfitting so. Hobbits! No report that I have heard does justice to the truth.’
Merry bowed; and Pippin got up and bowed low. ‘You are gracious, lord; or I hope that I may so take your words,’ he said. ‘And here is another marvel! I have wandered in many lands, since I left my home, and never till now have I found people that knew any story concerning hobbits.’
‘My people came out of the North long ago,’ said Théoden. ‘But I will not deceive you: we know no tales about hobbits. All that is said among us is that far away, over many hills and rivers, live the halfling folk that dwell in holes in sand-dunes. But there are no legends of their deeds, for it is said that they do little, and avoid the sight of men, being able to vanish in a twinkling; and they can change their voices to resemble the piping of birds. But it seems that more could be said.’
The Lord of the Rings Book Three, Chapter 9: Flotsam and Jetsam
Page 557-8 (Houghton Mifflin Harcourt; Single Volume 50th Anniversary Edition)
Dwarves have a much longer history and while they use Westron to communicate with other people, they have their own language that they still use among themselves.
But in the Third Age close friendship still was found in many places between Men and Dwarves; and it was according to the nature of the Dwarves that, travelling and labouring and trading about the lands, as they did after the destruction of their ancient mansions, they should use the languages of Men among whom they dwelt. Yet in secret (a secret which unlike the Elves, they did not willingly unlock, even to their friends) they used their own strange tongue, changed little by the years; for it had become a tongue of lore rather than a cradle-speech, and they tended it and guarded it as a treasure of the past. Few of other race have succeeded in learning it. In this history it appears only in such place-names as Gimli revealed to his companions; and in the battle-cry which he uttered in the siege of the Hornburg. That at least was not secret, and had been heard on many a field since the world was young. Baruk Khazâd! Khazâd ai-mânu! ‘Axes of the Dwarves! The Dwarves are upon you!’
The Lord of the Rings Appendix F, Section I The Languages and Peoples of the Third Age
Page 1132-3 (Single volume 50th Anniversary Edition)