I am gladhappy to help with mysome suggestions .

This sentence is good, but it could sound more natural. One usually hears "happy to help" more often, though "glad to help" can certainly be said. Additionally, and it's difficult to explain why, but adding "some" before suggestions softens the phrase. Without it, "with my suggestions" almost sounds overconfident in what you are suggesting.

Secondly, I propose organizing training sessions to strengthen the team's skiills in performing their duties effeiciently.

Spelling errors: "team's skills" and "efficiently"

Finally, I would like to inform you that our monthly meeting has alreadybeen scheduled for next week, and we could announce these possible decisions to everyone if you wish.

Here, the word "already" is not necessary, and should instead be replaced with "been". Additionally, a comma is needed to separate the sentences.

EMAIL - MyAs requested, my suggestions for team productivity.

The original sentence is fine by grammar rules, but just comes across as a little unnatural in the word choice. I think that's because of the ambiguity between "my requested suggestions" meaning suggestions from you that were requested, or suggestions you requested from someone else.

You can clean up this ambiguity with different phrasing like suggested above.

I am glad to help with my suggestions .

Firstly, to improve the productivity in the office I recommend creating a reward incentive program for all employees who strive to achieve their goals during the month.

Secondly, I propose organizing training sessions to strengthen the team's skiills in performing their duties effeiciently.

Finally, I would like to inform you that our monthly meeting has already been scheduled for next week and we could announce these possible decisions to everyone if you wish.