bhadram karṇebhiḥ Ṛg veda 1.189.8 & svasti na indro RV 1.89.6

Verse 1: Ṛg veda 1.189.8 

A beautiful and well known mantra, which has been mistranslated with the best of intentions by countless online websites. For those who are keen to know the exact meaning supported by grammar, here it is. This mantra is recited before some atharvan upanishads, for instance praśna and gaṇeśa-atharva-śīrṣa. It appears at Taittirīya-Āraṇyaka 1.21.3 and is chanted in countless situations as a śānti mantra, closing with the now familiar ॐ शान्तिः शान्तिः शान्तिः 

ॐ भद्रं कर्णेभिः शृणुयाम देवाः ।

भद्रं पश्येमाक्षभिर्यजत्राः ।

स्थिरैरङ्गैस्तुष्टुवाग्‍ँसस्तनूभिः ।

व्यशेम देवहितं यदायुः ।

Translation

O devas, may we (always/only) hear what is auspicious with our ears,
O gods/worthy of sacrifice, may we (always/only) see what is auspicious with our eyes; O gods/who are praised, may we reach/attain with steady limbs and bodies the life(span) that has been established by the gods. 

padaccheda:

भद्रं कर्णेभिः शृणुयाम देवाः ।
भद्रं पश्येम अक्षभिः यजत्राः।
स्थिरैः अङ्गैः तुष्टुवाँसः तनुभिः ।
व्यशेम देवहितं यदायुः ।

anvaya:

देवाः, कर्णेभिः भद्रं  शृणुयाम । यजत्राः, अक्षभिः भद्रं पश्येम । तुष्टुवाँसः, स्थिरैः अङ्गैः तनुभिः (च ) यद् आयुः देवहितम् (अस्ति, तद् )व्यशेम ।

padaparicaya:

भद्रं - auspicious, excellent, good; neut, acc, sing
कर्णेभिः* - with the ears; masc, inst, pl 
शृणुयाम - may we hear; 1st person, plural, optative, √śru

देवाः - (O you) gods, shining ones; masc, vocative, plural
भद्रं - auspicious, excellent, good; neut, acc, sing
पश्येम - may we see; 1st person, plural, optative, √paś

अक्षभिः - with the eyes; neut, inst, pl
यजत्राः - (O you) worthy of worship, worthy of sacrifice; masc, vocative, pl
स्थिरैः - with steady - neut, inst, pl, adj
अङ्गैः - with limbs - neut, inst, pl
तुष्टुवाँसः - (O you who are) praised; masc, vocative, plural, perfect active participle, √stu
तनुभिः - with bodies; fem, inst, plural
व्यशेम - may we attain, reach, consume, take possession of; 1st person, plural, optative, vi√aś

देव-हितम् - placed, put, established by the gods; neut, acc, sing, ppp/adj   
यदायुः (यद् आयुः) - the life(span); neut, acc, sing

(*कर्णेभिः this conjugated form is Vedic. We would normally use कर्णैः)

Verse 2  Ṛg veda 1.189.6

स्वस्ति न इन्द्रो वृद्धश्रवाः ।

स्वस्ति नः पूषा विश्ववेदाः ।

स्वस्ति नस्तार्क्ष्यो अरिष्टनेमिः ।

स्वस्ति नो बृहस्पतिर्दधातु ॥

(ॐ शान्तिः शान्तिः शान्तिः ॥)

padaccheda & anvaya

The verb दधातु applies to each god: 

इन्द्रः वृद्धश्रवाः नः स्वस्ति (दधातु)।
पूषा विश्ववेदाः नः स्वस्ति  (दधातु) ।
तार्क्ष्यः अरिष्टनेमिः नः स्वस्ति (दधातु) ।
बृहस्पतिः नः स्वस्ति दधातु ॥

(ॐ शान्तिः शान्तिः शान्तिः ॥ This does not appear in the Ṛg veda or Taittirīya-Āraṇyaka.)

Translation: May Indra (of) exalted fame give us well being; may the omniscient Pūṣā give us well being; may Tārkṣya of indestructible wheel rims give us well being; may Bṛhaspati give us well being. Om, peace, peace, peace.

इन्द्रः - Indra; masc, nom, sing
वृद्ध-श्रवाः - of augmented, exalted, wide fame; masc, nom, sing, adj
नः - to us; dative plural, 1st person pronoun; asmad
स्वस्ति - welfare, well being, prosperity, luck; fem, acc, sing
पूषा - Pūṣan, early vedic god; masc, nom, sing

विश्ववेदाः - all knowing, omniscient; masc, nom, sing, adj
नः - to us; dative plural, 1st person pronoun; asmad
स्वस्ति - welfare, well being, prosperity, luck; fem, acc, sing 
तार्क्ष्यः - mythical bird later associated with the sun and Garuḍa

अरिष्टनेमिः - the felly of whose wheel is un-damaged; epithet of Tārkṣya
नः -dative plural, 1st person pronoun; asmad

स्वस्ति - welfare, well being, prosperity, luck; fem, acc, sing

बृहस्पतिः - Lord of prayer, early Vedic sage; masc, nom, sing
नः - to us; dative plural, 1st person pronoun; asmad

स्वस्ति welfare, well being, prosperity, luck; fem, acc, sing

दधातु - may he give; 3rd, person, singular, imperative, √dhā

ॐ शान्तिः शान्तिः शान्तिः ॥

Rig Veda 1.1.1 (1st verse of the Hymn to Agni)

This hymn, attributed to the lineage of Vishvamitra and is in the Gayatri metre. It's probably the most translated and familiar Vedic hmyn, being the very 1st hymn in the oldest of the Vedas - the Rig Veda. Yet, as many  online sources as I have seen, there seem to be as many translations... As I said with 'Vande Mataram,' poetic licence is often taken while translating. Add to that the fact that Vedic Sanskrit differs from Classical Sanskrit! I'm sure there are much more beautiful translations you could find but here's what the grammar says:

अग्निमीळे पुरोहितं यज्ञस्य देवं ऋत्विजं  |

होतारं रत्नधातमम्  ||

vigraha from the padapāṭha:

अग्निम् ईळे  पुरोहितं यज्ञस्य देवं ऋत्विजं होतारं रत्नधातमम् 

I venerate Agni, the priest (lit. who is placed on the seat of honour/in the East), divine ministrant of the sacrifice, bestower of treasure, par excellence. 

अग्निम् - [अग्नि को ] accusative singular of Agni, Vedic god of fire. As you would expect, since the verb is in 1st person, 'I venerate', the rest of the verse is mostly in the accusative. agnim, hotaaram, ritvijam etc.

ईळे  - [I] venerate; 1st person singular from root ईड् (2A) - to implore, to extoll

पुरोहितम् - [पुरोहित को] priest, placed on the seat in the East [पुरः+हित] or placed ahead; accusative singular  

यज्ञस्य - of the sacrifice; genitive singular of यज्ञ masc. noun

देवम् - divine, if taken as an adjective; [देव को] if read as acc. singular of deva, masc. noun 

ऋत्विजम् - priest; accusative singular

होतारम् - priest who recites hymns of the Rig Veda

रत्नधातमम् - superlative of 'bestower of treasure/wealth' रत्न+धाता; accusative bahuvrihi compound, refering to Agni.

I shall continue with the translation of the rest of the hymn verse by verse. Feedback and comments welcome!