Should I let him cry it out?? How long should I let him cry it out?
Under no circumstances should a baby be left to CIO. BW does not in any way advocate this method as it breaks the bonds of trust between baby and mother.
At 7 weeks, I would strongly suggest feeding him if he wakes at 5am. In all honesty, any night waking after baby has slept for 3 hrs or more should be assumed to be hunger driven. He is still so tiny that he does need to eat very frequently.
There is new research that supports the use of a paci until 6 months of age as there is evidence that it reduces the risk of SIDS. Babies do have a very strong need to suck, a paci can be great for this!
Also does he need to be soothed before every nap and at nighttime?
Yes. He is only 7 weeks old.
Please try to remember that independent sleep is something that takes lots of time to teach to a baby. There will be months of night feedings and wake ups. This is all totally normal - not great for momma, but normal for any baby. At 7 weeks, this baby is still so new to the world that any sleep alone is a huge achievement...
Have you got a typical day's routine for me to look at? Could just be a case of needing a new routine...