Most software has some setting for the Thermal Mass of the mash tun, you can try changing that until the mash temperature matches your actual reading. You also want to make sure it's set up so that it accounts for the fact that you pre-heated the tun, otherwise it will tell you to make the water hot enough to heat the grain and the tun. Is it possible you just didn't wait long enough for the temperature to settle. That can take 5 minutes or more for the excess heat to get absorbed by the tun. Since you got the heat down to 152F and it finished at 149.9F, your 155F might have dropped down to 153F all on it's own, meaning you would have only been off by a trivial 1 degree.
I wouldn't worry about not hitting the mashout temperatures too much. A mashout doesn't serve much purpose to a batch sparger, frankly. However, you want to be able to hit your temperatures accurately if you ever want to try a step-mash. The program told you to add 3.25 (13 qts) gallons at 207F, though, but you only added 6.5 qts at 205F. On my system, depending on the mash volume and grain amount, that could account for a 10F difference in mashout temperature. That does sound like it was supposed to be your sparge water, though, unless you were intentionally doing a no-sparge batch.
A 7.5 gallon kettle is tight for 5 gallons, but do-able. Tubercle has a good suggestion, but it also seems like you have high evaporation at 2.25 gallons. You may be able to turn the heat down, get a less aggressive boil, and lose less to evaporation. I have tempered my evaporation to 1.5 gallons in my 8 gallon kettle, so that I could start at 6.5 gallons and end at 5 gallons, which would give you a gallon of headroom. You also could make a little less beer. There's no reason you can't design your recipes to make 4.75 gallons instead of 5, if that's what fits in your kettle. All that said, if you really want a new kettle, my buddy uses 9 gallon pots, starts with 7 gallons and has a comfortable amount of room to spare.