Istanbul long stopover with ITA matrix
 

Hi guys!
I am Not sure if this is the right forum to ask this question so my apology if it's not.

Well, I want to book a flight from London to Tel-Aviv via Istanbul with a long stopover (layover? not sure haha) of more than 48h.
I found the following flight: LON - IST (49h stopover) - TLV
Unfortunately, I am not allowed to add any attachments to show you the search but I used the "miconnect" function on ITA matrix for IST.

Now, I am not sure if I will be able to walk out of the airport, spend 2 days in Istanbul and continue to TLV, would I face any issues when I want to get out of the airport/get back to do check-in, or take my luggage with me when I am leaving the airport?

And in more general aspect, can I do this type of stopovers in ANY connecting city? let's say for example LON - NYC (48h stopover) - MIA? cause I can find it on ITA matrix but I am not sure if It's legal, and if the airlines will respect my booking.

Thanks!