not 100% sure, but based on what I have experienced, no
as long as you are still alive after placing your sentry gun, for example you get riley in five kills, as long as you stay alive riley's kills will count towards your next streak but if you die then riley can get as many kills he wants but it wont count towards your streaks
so like this
Riley called in next streak is sentry which is seven kills without hardline, you get a gun kill and then riley gets a kill you get a sentry gun
same case scenario but instead you die before riley gets the 7th killstreak for your sentry gun, then you dont get the sentry gun
same applies to all streaks, as long as you dont die once you have used them they will count towards your next streak
And, as the tips bit during loading screen says, anything you get from a Care Package won't count your streaks either.
Yes the ims and sentry do really well for linking scorestreaks, but they dont help your next cycle of scorestreaks! Once you start over they are just kills.
Yes they do count if you earned it directly (ie not from a care package) and you use it in the same life that you earned it.
To expand on this question, what about total kills as in the leaderboards longest killstreak, reason being I went 15 and 0 in one match, but it did not register the 15 kills, No care package was used, just normal killstreak rewards - IMS and Battle Hine I think were the ones I were running at that time.
The killstreak stat is generally just gun and lethal kills. So scorestreak kills don't count for it.