Personally, we did buy a seat for DD when we flew, even when she was "lap eligible".

In my mind, babies are the most fragile of all the travelers, besides older folk, and need the most protection. As such, I prefer to play it safe and buckle the little one in during the flight "just in case".
In addition, most kids of toddler and younger enjoy their own car seat, as it makes them more comfortable in a foreign space. This allows them to sleep more easily, and for you, you know it is fairly clean (at least clean enough for daily use).
That said, make sure the car seat used is also FAA rated (it will have a sticker saying this). I posted this on another thread, but I'll repeat it here, since this thread is applicable.
The FAA has guideline for car seats on aircraft. The pamphlet can be found here:
http://www.faa.gov/passengers/fly_children/
The governing pronouncement is Circular AC 120-87B (9/17/2012) found:
http://www.faa.gov/documentLibrary/media/Advisory_Circular/AC 120-87B.pdf
It is 14 pages, and definitely worth your time to look through it. Quite a few gate agents and some flight attendants don't know the rules, and make a judgement call regarding whether you need to gate check a car seat. Having this 14 page document printed will give you leverage in case they don't understand the rules.
In addition, you airline's Contract of Carriage will talk about accompanied minors and the airline's rules regarding them. Also worth a quick review.
In summary, please look at the FAA's rules regarding car seats on aircraft, if you use a car seat print a copy of the rules to have with you just in case (or keep a soft copy on an iPad), and please consider getting a seat for both kids on the plane.
As for the placement of car seats near a window, this is what the Circular says:
18. PLACEMENT OF CRS ON THE AIRCRAFT.
A window seat is the preferred location; however, other locations may be acceptable, provided the CRS does not block the egress of any passenger, including the childs parent or guardian, to the aisle used to evacuate the aircraft. Th e regulations contained in §§ 91.107, 121.311, 125.211, and
135.128 allow aircraft operators to determine the most appropriate passenger seat location for CRSs based on safe operating practices.
a. Aisle Seats.
CRSs should not be placed in an aisle seat because this placement has the highest risk of slowing down the passenger flow rate during an evacuation.
In summary, car seats can be placed in any seat, but the airline has the discretion to say where they may be allowed; however, the airline is not allowed to restrict their use, just move the family to a suitable place.
The reason for a window seat is that it doesn't restrict the others from moving out of a row. The same reason is why the circular suggests not using the rows in front of and behind an exit row. Just in case room needs to be made, but the circular does not restrict seats from being set here, it is up to the airline based on evacuation procedures.
Good luck and have a great trip!