βιόω ~ βιώσομαι ~ ἐβίων: live, pass one's life
φείδομαι ~ φείσομαι ~ ἐφεισάμην: spare, use sparingly
ἐλεέω ~ ἐλεήσω ~ ἠλέησα: feel pity, show mercy
πείθω ~ πείσω ~ ἔπεισα: persuade; trust, obey
κρίνω ~ κρινέω ~ ἔκρινα: decide, judge
These are the proverbs (and there are always more proverbs at the blog):
Ἀλύπως βίου.
Χρόνου φείδου.
Κύριε, ἐλέησον.
Μὴ κρίνετε, καὶ οὐ μὴ κριθῆτε.
Δῶρα πείθειν καὶ θεοὺς λόγος.
And now, some commentary:
Ἀλύπως βίου.
Live without grieving.
The word βίου is a verb here, a middle imperative of the contract verb βιόω. Yes, βίου is also the genitive singular of the noun βίος; the two words are from the same root, but they are not the same word. Just like in English, which abounds in homonyms (both homographs and homophones), there are some Greek words which can have the same spelling and pronunciation but which are different words, and you have to figure it out from context. In this context, βίου as noun does not make sense, but βίου as verb does. More about the many different types of homonyms at Wikipedia.
Χρόνου φείδου.
Be sparing of time.
The verb φείδου is another middle imperative; the verb φείδομαι has only middle forms, and it takes a genitive complement: χρόνου. This and the previous saying are from the maxims that Stobaeus attributed to the Seven Sages; see Wikipedia: Delphic maxims for more.
Κύριε, ἐλέησον.
Lord, have mercy.
The word ἐλέησον is a different kind of imperative; this is an aorist imperative from the verb ἐλεέω. The sigma is a clue that you are dealing with an aorist imperative; like most contract verbs, ἐλεέω has a sigmatic aorist: ἠλέησα (stem: ἐλέησ-). This phrase from Christian Greek is also found in Christian Latin: Kyrie eleison, or simply Kyrie. You can find out more at Wikipedia: Kyrie.
Μὴ κρίνετε, καὶ οὐ μὴ κριθῆτε.
Do not judge, and you will not be judged.
The words are from the Gospel of Luke. The full verse reads: καὶ μὴ κρίνετε, καὶ οὐ μὴ κριθῆτε· καὶ μὴ καταδικάζετε, καὶ οὐ μὴ καταδικασθῆτε. ἀπολύετε, καὶ ἀπολυθήσεσθε· — translated in the King James version as: "Judge not, and ye shall not be judged: condemn not, and ye shall not be condemned: forgive, and ye shall be forgiven." Note the emphatic use of οὐ μή for the negative prediction; more about Greek οὐ μή.
Δῶρα πείθειν καὶ θεοὺς λόγος.
The story goes that gifts persuade even the gods.
Note the adverbial use of καί here: even the gods, i.e. [mortals] and gods. The noun λόγος here means something like saying or story, or even simply "it is said," and it introduces indirect statement, with the infinitive πείθειν and an accusative subject, δῶρα. The line comes from Euripides' Medea, when Medea is handing over to Jason the deadly dowry for his new wife, so Medea is actually using this λόγος to lead him into her trap.
And here's a random proverb too: