Not if it's a personal residence. Entering someone else's property is trespass unless you have license. When private property is open to the public, there is an implied invitation to the public to enter, so you have license to do so unless it's revoked. With a personal residence, however, there is no implied license for strangers to enter (though there might be based on the parties' relationships or prior dealings).

There are some states with specific rules for personal residence, but it's inconsistent.

I don't think this is true. "Trespass is defined by the act of knowingly entering another person’s property without permission."

https://www.law.cornell.edu/wex/trespass

So it's illegal to, for example, go door to door looking for one that somebody forgot to lock and then spend the night there.

Under UK law, trespassing is a civil not criminal matter and so by some definition it is not illegal.

Still illegal, just not a crime.

No, it's unlawful, not illegal.

Definition of illegal: not according to or authorized by law : unlawful, illicit; also : not sanctioned by official rules (as of a game)

(https://www.merriam-webster.com/dictionary/illegal)

In legal terms, illegal and unlawful are not synonymous. In the UK, trespass is only illegal in certain circumstances: https://cps.gov.uk/legal-guidance/trespass-and-nuisance-land

Beyond that, it's only unlawful.

Where I live it is 100% legal to shoot them with no questions asked.

100% legal (castle doctrine) to shoot them, think about that for a minute, not generally legal to shoot someone engaging in a legal activity.

--edit--

Also legal to shoot them through the door but probably not such a good plan...

No, you just think it is. The intruder must be there to commit a further crime, usually a violent one.

   An intruder must be making (or have made) an attempt to unlawfully or forcibly enter an occupied 
     residence, business, or vehicle.
   The intruder must be acting unlawfully (the castle doctrine does not allow a right to use force 
     against officers of the law, acting in the course of their legal duties).
   The occupant(s) of the home must reasonably believe the intruder intends to inflict serious bodily 
     harm or death upon an occupant of the home. Some states apply the Castle Doctrine if the occupant(s) of the home 
     reasonably believe the intruder intends to commit a lesser felony such as arson or burglary.
   The occupant(s) of the home must not have provoked or instigated an intrusion; or, provoked/instigated 
     an intruder's threat or use of deadly force.

https://en.wikipedia.org/wiki/Castle_doctrine#Conditions_of_...

No, you must reasonably believe they intend to commit a further crime.

Here, unless you give them a reason they're just like "yeah, dude opened the wrong door, heh?"

I am aware of a few southern states where the statute assumes the intruder intends to harm you. UncleEntity is likely correct.