Languages understandable to English-speakers without learning

Long story short: no. Most native speakers of English who have no experience in any foreign language do not get much out of overhearing conversations in other languages.

David Ives, via the play "Universal Language," (an artificial koiné for theatrical entertainment) might argue with me on this one, but with English being the bastard child of a language that it is, most native English speakers who have no experience with any foreign languages are not able to pick up more than a vague meaning when hearing unfamiliar languages, even those within the same "language family."

You would think, for example, that conversational German would be somewhat (can't say easily) accessible to a native English speaker, but aside from conversational pleasantries afforded us all by UG, most anglophones do not understand German. Despite the heavy French influence, most anglophones do not understand French, either, unless they study it or have some other intense exposure.

What you've described is an excellent question from a linguistics standpoint. For example, I speak French, and I can understand the other Romance languages that I have not studied. It is interesting to note that native Spanish speakers often have trouble understanding conversational Portuguese, but native Portuguese speakers can work with conversational Spanish.

It does depend on your definition of "language" in relation to dialects. There are many dialects that most native English speakers can understand, and there are also quite a few that are unintelligible depending on the region. Tangier Island is a great example of a remarkable preservation of the English of the Colonies.

As near as I can tell, English is different than a lot of European languages in that scholars are much more inclusive. Where there are languages that other Germanic or Romance language speakers might call "separate languages", we would call "dialects". In other words, if two people can converse with each other in their own languages, and one is speaking English, then by definition they are both speaking English.

Not only that, but some dialects that aren't particularly mutually-intelligable are considered English. For instance, the first time I encountered full-blown AAVE, I couldn't understand a word. After years of heavy exposure I can mostly understand it now, but I can't speak it well. There's a joke about this in the movie Airplane!, where it is referred to as "Jive". I recently had the same experience when two Jamaicans started talking to each other (although they could have dropped into Patois. I'd have no way of knowing from one exposure)

I think the difference has to do with Nationalism. For example, Portugal and Spain have been separate countries for most of their existence. So as a matter of national pride, it simply wouldn't do for the Portuguese to refer to their tongue as "a dialect of Spanish", even if technically that's what it really should be. Similarly, Castilian speakers have been running Spain for a long time, and would not put up with people teaching their kids they speak some dialect called "Southern Castillian". They speak Spanish.

I submit that the same is probably going on with a lot of dialects of closely-related slavic "languages" that are associated with (actual or aspirational) national boundries. In fact, the accepted answer on How are languages and dialects distinguished from each other? over on Linguistics.SE makes this exact claim.

The Scots language is most likely the closest to English, whether it's a distinct language or a just a dialect of English is controversial.
Here's a sample of Doric Scots: How to speak Scots Doric [youtube].

Many scholars have studied the inter-dialect (language?) understanding of speakers of Swedish, Danish, and Norwegian. At recent meeting I attended, all Swedes, Danes, and Norwegians spoke their mother tongue and were understood. Einar Haugen, the great linguist, described the situation as Semicommunication: The Language Gap in Scandinavia. Non-specialist's understanding of neighbor-languages differs among the triad, and it turns out Norwegians are better at understanding Danish and Swedish than any other dyad. This may have something to do with attitudes (Norwegians grow up with hundred of dialects -- not to speak of two standard written languages -- bokmål and nynorsk.

I understand I am supposed to avoid asking for help, but I am trying to develop a method to measure the distance between languages (or dialects).

No, not in the same sense as Spanish/Portuguese or Scandinavian languages.

You can often recognise some words in Danish/Dutch, especially if you have a northern English accent and it's fairly easy to follow the structure of a simple sentence in German (especially because most of the question words are similar)

You can make out some French vocabulary because of the same Latin origin.