Unfamiliar metaphors in Hosea 14:6-7

“1 ¶ O Israel, return unto the LORD thy God; for thou hast fallen by thine iniquity. 2 Take with you words, and turn to the LORD: say unto him, Take away all iniquity, and receive [us] graciously: so will we render the calves of our lips.

render the calves of our lips.

Make recompense for having made our own sacrifices or "kissing the calf".See here

Asshur shall not save us;

The pun ish-shuwr as 'fire wall' or 'wall of fire' such as protected Israel when fleeing Egypt. Ex 14.20

   Ex 14:20 And it came between the camp of the Egyptians and the camp of Israel; and it was a cloud and darkness [to them], but it gave light by night [to these]: so that the one came not near the other all the night.

we will not ride upon horses:

We will not trust in earthly strength (De 17.16)

neither will we say any more to the work of our hands, [Ye are] our gods:

Refer to the answer concerning hewing stone hewing stone. In the act of chiseling stone we add our works which we then worship. God tells us to use unhewn stones for the altars.

for in thee the fatherless findeth mercy. 4 ¶ I will heal their backsliding, I will love them freely: for mine anger is turned away from him.

Him....notice the change to the first person singular as a reference to Christ.

5 I will be as the dew unto Israel:

Just a taste of the living water in the morning hours. After the cross Israel only got a taste of the living water which will be given to overflowing in the future.

he shall grow as the lily,

Shushan means 'whiteness' which is the color of covered sin. The lily grows "between the waters" like the firmament. He will reconcile Holiness and Grace covering our sin in the process.

and cast forth his roots as Lebanon.

"Smite his roots as whiteness" Lebanon also means 'whiteness'. The tree of Lenanon is Christ, his root, entwined in the earth is his incarnation. His flesh was smitten though 'he knew no sin'.

6 His branches shall spread,

He is the vine, we are the branches. Joh 15:5

and his beauty shall be as the olive tree,

olive (from an unused root akin to ziv or 'brightness'. Beauty and brightness are associated with wisdom (Ez 28.7, 17)

It is suggestive of the tree of knowledge. The purpose of the tree of knowledge was to teach Adam of God's holiness by a symbol of separation. When he violated the separation, he had to be physically separated by removal from the garden. The opposite example from Adam's transgression is Jesus in the kenosis (Ph 2.4-7)

and his smell as Lebanon.

His smell as whiteness. Christ offer perfect obedience to the Father which is represented by the burnt offering. It is a 'sweet savor unto the Lord'.

7 They that dwell under his shadow shall return; they shall revive [as] the corn, and grow as the vine: the scent thereof [shall be] as the wine of Lebanon.

Shadow is grace. The corn has to die to bring forth life. Israel will be reborn and be grafted back on to the vine.

Wine represents grace (strong drink represents the law). Those who had departed will be restored. Their reconciliation will have the same sweet savor as his total devotion, being the result of his grace.