Quick pay off the student loans. You have 140K in savings with a combined salary of 170K. You are looking to make money with the 140K, so just pay off the loans. It will turn your monthly loan payments in to a stream of money that can be used to save money for the next house.
The rest of the 700K needed for a 1.5 Million dollar house has to come from savings and the profit from selling the first house. If the house sells for loan balance +300K you still need 400K in savings.
Turning 140K into 400K in 5 years will funneling a large amount of your income into savings or excellent returns. Of course there is no way to predict return or what will happen to the market.
If you don't sell the first home, you can rent the house. You either hope that the rental you charge allows you a positive cash flow. Or you hope that the house appreciates in value, so you hold on to it even if the rental income is a little below break even. Of course some keep the house because they can't sell it.
In your case the equity might be more important for you to purchase the next house.
A friend tweeted a similar question regarding student loans, and I responded with Student Loans and Your First Mortgage.
The punchline is that you need to be aware of the 28/36 ratios in a bank qualifying you for your mortgage. Even though you have a house, you may not be aware of this.
Simply put, 28% of gross monthly income can be used to qualify for your house burden, loan, taxes, etc. 36% for total debt. So the student loan may fit in that 8% gap, and paying it all off reduces the cash you have without helping you borrow more money.
3-5 years is short term, and to that part of the question, this money should not be invested in anything at risk. A 3 year treasury or CD would be it, in my opinion.
To your secondary question:
Appropriately consider all estimated numbers involved with keeping the house compared to your closest estimate of what the home could sell for. Weigh out the pros and cons yourself as a stranger will not be able to 100% appreciate what you value and dislike. Remember to include insurances, taxes, HOA(s), and the actual mortgage payment. Depending on how you also plan to rent out the property, include whichever utilities you intend to cover (if any). There will also be costs for property management and upkeep as things will break overtime and tenants will not hesitate to get you (or your management) to fix them, either way that means you are paying.
I would also keep in mind while homes typically appreciate in value there is a higher risk with tenants for the value to depreciate to damages and poor upkeep. There are increased legal risks to renting, so be sure you have properly vetted whichever management you are going with. In extreme circumstances you also could be required to retain an attorney to defend yourself again litigation because whichever management team you hire will most likely defend themselves and not include you in that umbrella. My family lives in the LA area as well and a judge refused to throw out an obvious frivolous suit when my parents attempted to rent out a house. The possible renters after signing the main paperwork never showed to finish a second set of documents for renting. Parents immediately declined to rent to these people as they missed something so important without any explanation and they sued claiming racism, emotional damages, and some other really crazy things despite my parents never having met them (first meeting was between property management and renters only).
Personally and professionally, I would only suggest renting our the place and not selling if you can turn a profit after all the above mentioned costs. If renters are only paying to keep the property in the black you have yourself a non-earning asset which WILL be damaged over time and require repairs which will come out of your pocket. Also, while the property is unoccupied you also must remember it is not earning at that time.
Much of this may sound obvious, overcautious, etc... I simply wish to provide my family's experience to help you in making your decisions.
Best of luck with your endeavor.
Edit: Also, you will be required to report all earned rental income on your taxes. They will fall under the Schedule E and possibly K-1 area. I would strongly recommend consulting with an actual accountant about the impacts to you.