There is nothing you can do right now but put your blind trust on your husband. Your worries will only fuel bigger issues in the future. He is doing his best to keep you on the loop by talking to you often

"we talk maybe 45 minutes a day after he returns home from work and before he heads to word and sometimes on his days off yet I feel he has more free time"

I understand that his past history has made you insecure thats why you felt that the time he is giving isn't enough and everything else is still questionable. This is not very healthy if you wish to keep your marriage.  I have been in a long distance marriage with my husband before I moved here in Qatar and yes there are some things that I question too, but I try not to bother myself about it. I've come to accept that he is far and yes there are alot of possibilities that can happen however, our relationship is more important to me than these petty issues I've created in my head. As long as he is doing his part as a husband, as long as the communication is still there, as long as you can still feel his efforts, it is worth remembering that he chose to be far to give you and your family a better life..but if you really are bothered about your worries, your gut feeling is giving you sleepless nights by all means give him a surprise visit in Qatar.