Which of the following is more appropriate / polite?

I would like to bring a couple of facts (or things?) to your notice.
OR
I would like to let you know a couple of facts.

Please advise.


Question: Proper word for "understood"
Answer: Google suggests many, as well as a few of my own:


Clear
Comprehensible
Transparent (Although this is usually said as an insult to an individual who has a plan, so I wouldn't recommend it)
Distinct
Concise


Although these are all options, yours fits the bill just fine, if not a little formal. For a word in a casual conversation or chat1 you might like to try:


  "Yeah, I get you"
  
  "It's all clear"


In summary, you've used the right word, but there are some good alternatives out there if you wish to use them!


1: Although that's not to say that "understood" would be completely inappropriate in a casual conversation!

EDIT:

To express sympathy, you could (and should) use different terms to those that describe a knowledgeable understanding. If the person is going through a bad time:


"I feel for you" (when the person is going through a bad time)
"I know how you feel"


Or, if the person is judging their actions (or themselves) in response to their situation and circumstances:


"You did the right thing"
"It's not your fault"
"You did the best you could"


Or, if you've been through a similar experience:


"I've been there"
"We've all been there" (if her situation is a common one)