Honestly not surprised. You're running that on a, what, 15 year old potato? Yes, Skyrim isn't the most modern game, but that GPU was intended for desktop use, back then. I'm actually impressed that it runs at all. I'd half expect the game to just crash because it requests something that dinky GPU simply cannot provide.

HD graphics 4000 can barely run Minecraft let alone skyrim. I think you're getting what you're getting.
Crank literally everything to minimum.

Its technically running.

If you really want to make Skyrim playable on an HD Graphics card, you might have to go lower than even the low settings (like by editing config files, for example).

Many people have made recommendations on what to change in videos and posts, you should look them up. IIRC there was even a mod for low-spec Skyrim.

This is really a hardware limitation. If you are in a desktop you should add a graphics to it. But if it is a laptop unfortunately there is nothing you can do.

Really the only things I'm seeing in the settings that could get you a few frames are lowering the resolution, and turning shadows down further. Beyond that, the answer is "buy a new computer". The i7 3770 is by no means an impressive processor, and you're putting double duty on it since you don't have a dedicated graphics card.

You're also underclocked compared to the published specs, so I assume you're in a laptop where heat is a major bottleneck as well?

Disable ssao and reduce shadow distance. Also get a potato mod. You might find skyrim legendary edition to be more bearable fyi

I ended up installing the original Skyrim, and it works fine. All hail the mighty Razor 1911.