Hi Skullcap,
Yes,HMRC will be informed of your pension details.
You can check your tax code online if you have a personal tax account here:
Personal tax account: sign in or set up
If you are unsure if your tax code is correct you can also contact HMRC to review:
Income Tax: general enquiries
No, you would not need to complete a Self Assessment tax return.
You can check if you meet the criteria for Self Assessment here:
Check if you need to send a Self Assessment tax return
At the tax year end HMRC will review your record and if tax is underpaid we will send a calculation to you.
Thank you.