Not familiar with that engine, but it looks like a 2 cycle.
Believe it or not Briggs made a 2 cycle and they had a thermostatic compression release.
By looking at the side the round cover may be a compression release. All the symptoms you stated can be a compression release problem.

The carb is the part featured in your first pic w/ a round "bowl" on the bottom. Coming this far, you will want to at least want to ensure the fuel system is clean as well. Removing the nut in the center of the bowl will allow you do do a basic cleaning.It may be easier removing the lower plastic shroud to service the carb and get spec #s.

Here is a link to the service manual:

Here is a link that might be useful: 2cycle manual

Ha! Getting the rest of the shroud off will take some time for me. Also, the bowl...there is a nut on the bottom...I feel nothing with my finger. I found my snowthrower manual...its a 1996 single stage 2 cycle Simplicity with the Tecumseh engine (HSK600 Snow King Hsk845 or HSK850)

What is a compression related promblem and is something worth fixing...and how should I try to fix it? I AM A COMPLETE NOVICE. I was proud just to get the part of the shroud off. I am not shy about getting my hands dirty though!

Note the manual has instructions for storage that says to drain the carb (if so equipped) by pressing up on the bowl drain..not really sure what to do here. The pic shows the bowl drain as beinga pertruding dohicky on thebottom of the bow. It also shows a round screw. Mine feels smooth on the bottom unless I am feeling the wrong part.

The bowl drains became a thing of the past. Maybe a better option would be to mark the hole the linkage goes into and just remove the carb. I've been out of the snow belt for a while. I remember some reed valve issues (compression release) on the real old ones. I don't know how likely it is that yours has an issue. If the reed isn't broken, then it is merely a clearance issue. Maybe someone else can chime in on the likelyhood. The different styles are covered in the manual.

ok, that last post lost me....I am a complete novice. Reference a part of the pic and tell me what I should do and I am willing to give it a whirl...