I've just booked five sectors with DL (six actually because DTW - BWI - JFK with a 1 hr 50 min stopover was over $300 cheaper than DTW - JFK/LGA direct) and booked it as two tickets as it was cheaper. DL appears to use the philosophy used by LoCo carriers with single sector pricing but their website struggles if some flights can be booked in the lowest fare category but others at a reasonable time only only available in slightly less discounted bands). This will work where you are stopping over (I am spending three nights in Chicago between my tickets), DON'T try it if you are trying a quick connection.
As for through booking, by all means try it if you are on different tickets but don't bank on the airlines playing along, especially if the carriers are in different alliances. Even then there have been several posts in PPRuNe where BMI wouldn't interline to other Star Alliance carriers at LHR. Many years ago it was the norm, I took an early morning DL flight from MCO - JFK so I could spend the afternoon in NY before departing with BA to LHR. The DL agent checked the details on my BA ticket typed in a few details and hey presto! I doubt that would happen today, but then again fares are much lower.
Incidently, it is often worth trying a consolidator such as Trailfinders. A couple of years ago I booked a six sector itinerary with BA / QF. You couldn't (at least then) book more than four sectors on the BA website. Rang up BA - would have incurred a booking charge. I then rang up Trailfinders who by shifting the itinerary by one day saved me £200. They may have contacts with different airlines and can book you on one ticket.
One other thing whilst I think of it. Different airlines / routes sometimes have different baggage allowances with a lower or no allowance on the shorter sectors. With a multi-sector itinerary you will generally the most generous allowance applied to all sectors.