Ok, let me try, but i think a native might explain it differently.
Yes connected is mostly physical, but not necessarily, like:
Don't mess with him, he is connected. Or he has connections he can get it done.
Here it means, he has links/relations with people in important positions.
Relations could be used in family ties, or quantities, like hike in food prices is related to increase in fuel prices.
And if you are into softwares and stuff, you will come across relationships, or relations in databases.
"He is involved in some crazy stuff." does not necessarily means he is forced to do things.
If whatever you are talking about is a circle to be involved means you are somewhere near or around it's center.
To be engaged could means many things, like when a man and woman get "engaged" means they are committing to get married.
To be engaged is also to be busy in something.
I don't see the forced or not forced situation in the use of these two words, but then again i am not the authority.
I hope my answer makes sense.