First of all hello everyone, thank you for sharing your knowledge and ideas here.

I recently bought e350 coupe from 2014.

Everything seems to be working fine but I noticed two distinctive sounds from under the hood. I would appreciate you help in locating the source of it.
I assume the humm might be related to LED lamps. Correct me if I'm wrong.
Other, more scary sounding noise is that squeaky buzz. Do you have any idea?
I attach the YouTube video I have taken.
Thanks in advance
Gregory

 

Well this is funny. I just bought an E350 2014 convertible like 3 weeks ago and when I was under the hood it definitely was not that loud. I could barely hear it, but there's a slight vibration over the headlamp. I googled it and yes they are fans to cool the LED headlights. Yours sound really loud so I don't know if they're on the way out. When you lock the car, if the headlamps aren't hot it should shut off in about 10 seconds. If you don't lock the car for some weird reason, they'll stay on for like 45 minutes or something crazy like that.

 

... have you noticed the HL fans running in the morning with HL: OFF + cold engine?

That makes you wonder what's going on...

This happens when chassis is still baking hot when SAM power times out. In the morning HL-Fans resume what they were doing.

There is a combination of two stock issues here.
-- HL-Fans cool HL with hot engine bay air that's hotter than HL... Fans actually heat up HL not cool it!

-- This evidence the engine bay accumulate extreme heat.

When heat is circulated out of engine, the LED fans stop 5 secs after HL and fuel rail does not hiss without stock heat soaks.

 

To be honest I just got the car. We drive it on the weekends but no I have not noticed the fans being on for the headlights in the morning. It's just been when I unlocked the car they immediately come on. If I don't lock the car they'll stay on for 45 minutes or so. If you lock the car within 10 seconds they shut off.

 

To drive on the weekend you may want to use a smart float charger (CTEK/NOCO) to safe guard your battery charge.

AGM's can last over 10 years in top shape or far less when drained down.

 

That sounds weird, and loud. Mine makes a little noise like some teeny little motor is running but I don't notice it unless the hood is open. I live where it's pretty much dead silent. If I were in normal ambient noise neighborhood I'm not sure I'd hear it at all.
That other noise at the 6 sec mark I've never heard before. Sounds like an original series Star Trek sound effect on the bridge. I think I'd have to track that down just to find out wth could possibly make such a noise.

My little elect motor sound happens regardless of headlight use, and I've never noticed any new sounds when I did use the headlights. I half-*** looked for my sound but it's so quiet and turns off so quickly I never did figure out where it's coming from. Mostly I just don't care enough to look any harder.

I'll second the point Cali made about being wary of battery drain. You didn't mention if it does this all the time, but my car did. It would only do it now and then, and turn off quickly after, but it turned out something was wrong and my ECU was On 24/7, for years. I only noticed it when the battery was aging and it was enough to kill it. I first noticed it when parked for 3 or 4 days and it was dead. I thought maybe I left a something on? Then after a few month it died again. Then it would die if parked for two days, then one day. I bought a new batt but never did figure out why this was happening. Then one day my Keyless Go stopped working and the problem went away. I got the Keyless Go to work again for a few days and the problem returned with it. Then it died for good, along with the problem.
Now the quiet little motor is only on for a short time after I turn the eng off. If I had to guess I suppose 20 seconds?

While my example was extreme, these cars can still kill a batt pretty quick. It seems to vary but some people here claim 2-3 weeks can kill it. Other don't seem to have a much of a problem but none of them are like the old days (before computers) where you could leave it sit for a year and it fires right up.
I'd never buy one of those expensive battery chargers, but it's nice to have something. I broke down and bought a cheapo charger when I had the aforementioned issue, which was $20 on Amazon.
I cut the clamps off it and added ~8' of wire with a quick connect plug. Now I don't need to bring the charger to the car, or open the hood or any of that bs. I simply plug it in. The connector on the car side I put at the drivers side bottom corner of the windshield, hidden by the hood when not used. Super convenient, if that sounds interesting to you.