Will dropping linear algebra negatively effect me in any way? Namely, would dropping it mean I'm not taking the required number of points in first year?
I'd keep it for the first few weeks of semester and then decide whether or not to drop it then. As long as you withdraw before the census date (generally after week 3 or 4 of a semester) you don't get a "withdrawn mark" on your transcript or have to pay any fee.
Unless you're working solid hours, managing a full-time load shouldn't be too difficult, especially in semester two when you should be well into the swing of things.
That said, I ended up doing only 3 subjects in my first semester largely unintentionally. I'm doing Civil Systems via the Environments course and picked Virtual Environments, a former first year subject (now second) known to many as one of the most full-on at the university in terms of workload. I ended dropping this before the census date, but after the cut off where you could enrol for a new subject, so ended up with only three subjects.
So then I had to pick up one in the summer to make up for it. I had done Calc 1 in 1st Sem and Calc 2 in 2nd, so I chose to do Linear Algebra over summer for precisely the reason that Hancock mentioned above: so I could do Eng Maths in Semester 1.
In the end, this was a really good decision. I found the intensive summer schedule to be advantageous for my style of learning, and the summer lecturer, Alex Ghitza, was fantastic. I ended up getting a mark of 87 for Linear Algebra compared to an average of 66 for Calc 1, Calc 2 and Eng Maths collectively.
So, overall I would recommend at least attempting the full load in semester 2 and preferably sticking to it, yet don't be scared of leaving it for summer if you don't mind the 15 days of solid work you'll need to do when you could otherwise be at the beach.
Have I included all the compulsory subjects for a civil systems major?
I think you're missing Structural Theory and Design, a 3rd year subject.