TL;DR Summary:
Yes, there have been some DC/Marvel crossovers that have taken place in an alternate universe where the characters share a single world; however, there have also been some that have not - including the one your image references.
As other answers have noted, there are a number of Marvel/DC crossovers where the characters appear to reside on the same Earth, starting with the Superman/Spider-Man "treasury" edition sized comic published in the mid-1970s.
Marvel and DC crossed over several more times in the late 1970s and early 1980s with the same assumptions (everyone on the same Earth): another Spider-Man/Superman story; Batman/Hulk; and X-Men/New Teen Titans.
These original 4 crossovers seem to be highly negotiated; they feature characters of comparable importance for both companies, both as the protagonists and the antagonists (first Supes/Spidey had Doctor Octopus and Luthor as the villains; X-Men/Titans has Darkseid and Dark Phoenix), where the story seemed to come from the characters that were negotiated, rather than the story idea coming first and the characters coming after (note that the Titans hadn't had anything to do with Darkseid, as I recall; Trigon would have made more sense). I'm not necessarily saying that's true - but, these certainly had that feel.
A fifth such story was planned for the mid 1980s (JLA/Avengers), but the negotiations came to an impasse, and it never materialized. DC/Marvel crossovers stopped for several years, at that point.
In the 1990s, as Dark Horse and Image became significant players in the marketplace, inter-company crossovers became more prevalent (not common, but not something with the earth-shaking significance with which fandom greeted the first Spidey/Superman crossover, or the X-Men/Titans one).
DC and Marvel did several crossovers during this time (Batman seemed to be the most likely participant from the DC side - I seem to recall crossovers with him and Captain America, and with him and the Punisher). However, the stories began to feel more natural, and less like a legal negotiation. I think even then most (but not necessarily all) of these stories involved the characters simply existing on the same Earth, rather than having to travel to some other dimension to meet.
In the late 1990s, Marvel and DC had a company-wide crossover: DC vs. Marvel/Marvel vs. DC. It was a 4 issue mini-series, and the order of the company names depended on which company published each issue. In this series, the characters did not live on a single Earth - the DC and Marvel Universes were apparently those actually published at that point, and the character were brought together to fight each other, some from the DCU to the Marvel U, some the other direction.
Aside: at one point the universes were merged, and both DC and Marvel published a week's worth of comics from the merged universe (the Amalgam Universe), with characters and groups that were a deliberate mash-up (Super-Soldier (Captain America/Superman), Iron Lantern (Iron Man/Green Lantern), JLX (Justice League/X-Men), etc). This was well-received enough that the companies did it again for one week about a year later, mostly with different titles and featured characters.
And, a few years later, in issues cover-dated September 2003 through March 2004, there was the four-issue JLA/Avengers / Avengers/JLA (again, the order depending on which company published which issues). This was not a resurrection of the 1980s proposed project; but, was a replacement of sorts for it. As I recall, multiple well-placed people in both companies remembered the abandoned project, and the disappointment when it fell into Limbo. The then-current writer of the Avengers, Kurt Busiek, had also worked for DC, and had at least worked with the JLA characters (he wrote a Red Tornado mini-series in the 1980s), and he was picked to write the crossover.
As noted elsewhere, this is where the image in the question comes from.
By the 1970s, both DC and Marvel had established multiple alternate universes of some sort. DC had Earth-1, Earth-2, Earth-3, Earth-S, Earth-X, etc.; Marvel had an alternate universe where Reed Richards had become the Thing, and, in 1975, started publishing What If, featuring stories of universes where some facet of Marvel history was different (Spider-Man joined the Fantastic Four; Wolverine became a vampire; etc.). Characters from different alternate worlds met on a regular basis in DC's comics; it was much rarer for this to happen at Marvel, but we did get a glimpse of a brand-new alternate universe on a monthly basis.
However, for legal reasons, interactions between characters from different companies was much rarer. While (technically) both the Marvel and DC multiverses include/included a world with both DC and Marvel characters on it, only stories published jointly by both companies would every see that world. In practical terms, the DC multiverse consisted of universes where all the characters are owned by DC, or in the public domain, and the Marvel multiverse consists of universes where all the characters are public domain, or owned by Marvel. At some point, the term omniverse was created to deal with this reality. A multiverse was seen as all the universes that could regularly interact, belonging to a particular company; whereas the omniverse consisted of all possible multiverses: DC, Marvel, Disney; Tarzan, Lord of the Rings, Sherlock Holmes; Indiana Jones, Alien, Predator; Star Trek, Buffy, Farscape; and so many more.
So, you can think of DC and Marvel as being (essentially) independent multiverses, that are part of the single omniverse. And, think that it's easier to moved around and visit within your own multiverse; it takes more than one set of the "Powers That Be" to journey out of your multiverse to another one.
This is Flash batteling Quicksilver. I dont know where its from, but how could they ever meet, since Quicksilver is from marvel?
And why can wade wilson and slade wilson have the same surname? Because its the same person or are they brothers/cousins or any other relatives, or coincidence?
