Could you please supply me with a few sentences including the word brath (without any mutation) which best convey the most common meaning(s) associated with the word? GRMA!

Do you mean "bráth" or "brath"? They're different words.

I'm looking in FGB and can't find an example of either with no mutation whatsoever.

What are you trying to say?

Redwolf

There are quite a few expressions that are made with - ag brath. You can pick out some of them here -

www.breis.focloir.ie/ga/eid/brath

and

www.breis.focloir.ie/ga/fgb/brath

As Redwolf said, I'm assuming you mean the word brath and not bráth.

I hear the word brath being used quite a bit but I find it difficult to figure out its meanings looking in the dictionary.

Here are some example using the word:

Tá mé ag brath ar imeacht
Cad atá siad ag brath a dhéanamh anois
Bhí eagla le brath sa seomra
Dar liom go raibh imní le brath ina glór
Más féidir brath ar an méid a dúirt sí

It seems to me that "brath ar" means "depends on", like in this sentence:

Is air atá an tír ag brath: Its on him the whole country depends.

Would you agree that most uses of the word "brath" are in relation to "depends on". Or are there other meanings a beginner should consider learning?

If you check the links Franc gave above you will see a lot of examples.
Like a lot of words in Irish "brath" has different meaning depending on context. The context will usually be fairly obvious.

From Wingléacht -
NOUN
perception
feeling
spying
betrayal
expectation
intention
dependence

VERB
perceiving
feeling
noting
betraying
sensing
intending
expecting
hoping for
depending


Tá mé ag brath ar imeacht THINKING OF/CONSIDERING
Cad atá siad ag brath a dhéanamh anois CONSIDERING/INTENDING
Bhí eagla le brath sa seomra PERCIEVE/SENSING
Dar liom go raibh imní le brath ina glór PERSIEVE/SENSING
Más féidir brath ar an méid a dúirt sí DEPEND ON