Put simply, for this particular mission, at the beginning of each round (meaning it happens after the end of your turn + client turn), you do what the progress track is telling you (in the same order than written btw).
Question 1Yes, you draw 2 more cards from your deck (even if you did it earlier during the normal steps of your previous turn). One does not prevent the other.
Question 2Yes, you heal and buy before beginning the round (even if you did some purchases during the normal steps of your previous turn). Same thing that in the situation in question 1 then.
Question 3The Scene's ends are mentionned only because some upgrades are rebooted at each scene. Nothing else happens. It is written on the mission card : "There are no additional rewards for runners when a Scene ends". The obstacles stay. I would say that the new scene begins at the same time you begin this round, but I am not sure.
Question 4Yup, as soon as you survive until the very beginning of round 7, you win! So, basically, you play 6 full rounds. The mention of a 7th round is probably because some Crossfire effects are trigerred when the card is discarded (Timebomb for example).
Btw,
Extraction can be hard if your runners have no Karma upgrades. What I like to do sometimes when I play solo is to take 2 runners and play some Crossfire missions. That way, I can enjoy the game even if I not with my group.
You can also try the mission "Ambulators", it is meant for runners with no Karma (again, I would suggest to play solo with 2 runners) :
http://cdn.shadowruntabletop.com/wp-content/uploads/CAT27700_Ambulators%20Mission%20Card.pdf?4713c7. The link seems down for now however. In the meantime, more info and missions are found on the FAQ page :
http://www.shadowruntabletop.com/shadowrun-crossfire-game/.
Have fun and do not hesitate to post more questions!