Learn English – What’s the meaning and usage of “Let’s keep in touch”

Keep on is a phrasal verb taking a gerund clause as its complement; it means “continue [a course of action]”. It is an intensive version of plain keep in this sense.

Bob kept on interrupting me, so I couldn't finish. ... Bob continued to interrupt me
Laura kept on writing until the chapter was done. ... Laura continued to write

It often has the sense of “persevere”, as in the 60s-70s catchphrase

Keep on keeping on. ... Don't give up the struggle to achieve/survive

Keep ... as it is is not a fixed phrase but the transitive verb keep, meaning “maintain [something] [in a specified condition]” combined with the phrase as it is serving as the second complement with the meaning “in its present condition”.

Ryan wanted to change the second sentence, but we decided to keep it as it was.
Let's keep the schedule as it is for now. We can update it when we know more.

The definitions of lest:

1. so as to prevent any possibility that ⇒ he fled the country lest he be captured and imprisoned
2. (after verbs or phrases expressing fear, worry, anxiety, etc) for fear that; in case ⇒ he was alarmed lest she should find out

So doing some substitutions:

Hold fast to prevent any possibility that you should fall. (If you hold fast you shouldn't fall.)

To prevent any possibility that anyone should think it's strange, let me assure you that it is quite true. (My assurance that it is true should make it impossible for you to think it is strange.)