Sump output to Storm Drain, 135 feet away?

I bought my house a year ago and we have two sewer pipes exiting the house. One out the front and one out the side of the house. My sump pump goes out the side pipe. A while ago i looked at the storm grate and noticed a 6 inch pipe flowing into storm pit and the pipe pointed right towards my house, which i thought was connected to the sump. During a recent heavy rain, my sump was runny every 5 seconds. I went to the street and didnt see much water coming out into the storm sewer and was concerned. The city came out and sent a camera through the pipe from the street and found that the pipe was not connected to anything. I then had them do a dye test on my sump pit and determined it is flowing into the waste sewer, which is illegal and needs to be fixed.

I want to connect it to the storm sewer.

So my questions are the following.
When the 1 1/2 inch pipe exits the house, what size pipe should it transition to and how soon after existing the house should it connect to this pipe?

I want to connect to the storm sewer which is about 150 feet away. What type of pipe should i be using? Can i use Schedule 40 or do i need to use SDR 26/35?

Will the underground pipe freeze in the winter?

Can i tie gutter run off into this pipe as well?

Any general advice or things to look out for when i get started?

The size of the pipe transition would somewhat depend on what else you add to the sump output. For example, if you are allowed to channel your gutters to this drain, then you'd probably want to transition to the max size as soon at those were introduced. otherwise, you wouldn't need to do anything until you get near the storm drain. Depending on how deep the line is, it is no different than your normal sewer...it should be below frost line, and you shouldn't have to worry. The pipe DOES need proper pitch (slope) to it. That slope depends somewhat on the diameter, but for less than (I think) 4", it should be 1/4" per foot. Larger pipe can get away with slightly less slope, but it should still have negative slope to the drain point. Since it is liquid at the house, you have proper pitch, and it is deep enough, the flowing water won't freeze.