Last year was my first year housing martins (5th year trying) I have a T14 house which was designed by Andy Troyer. I also use several of his gourds. So far this year mine started showing up last Friday and I have 3 pair staying and a few other's stopping in. I am hoping to get 15 pair this year.
As you mentioned you must eliminate sparrows and starlings. The best thing to do is get some SREH (starling resistant entry hole) to go over your existing housing. There are a few different designs but the starling can't enter them. They way the sparrows are your only worry. I have the exact same sparrow trap that you have. They work good but they are hit and mis. I might go three or four weeks and get nothing, then get 10-15 in one week. So don't be disappointed if it doesn't work right away. If you can keep some alive in the trap they will help draw more in. The first is the hardest to catch.
Lastly unfortunately martins don't eat mosquito's. Martins feed at high altitude during the day time and mosquito's are lower if out at all during the day. By the time mosquito's get really active the martins are in for the night. There have been several feeding studies that show they never eat them. This is one of those tricks some companies use to draw some business. Like you said a little effort can go a long way. Keeping the other birds out and checking the nest about once a week will give you a heads up if there are problems. Last year I found two dead babies in one nest about a week old. That would have been a stinking rotten mess for the others had I not found them.
Here is a pick of my house after the gourd rack was added.
