Seeking advice on first VA backed home loan purchase

Hey, all.

Here’s a little background:

Location – Northern NJ,
Expecting first child,
Credit score 750 spouse 780,
Looking to buy within the next 6 months.
Thinking 500k but would like to be under 450k,
Have a vehicle loan currently (13000 left),
Looking to buy new vehicle with the arrival of newborn,
So around September/October for the new car,
November/December we’d like to have closed on a home.,
Gross income for both of us 165k.

So all we’ve done at this point is just look at Trulia and saved some homes we like.
Just not really sure about the process.
I’ve only looked at mortgage rates through Trulia and it’s looking like 3.5% on the lower end.

I just have a bunch of questions.
Will having my wife on the loan as a Co borrower be helpful?
Is the first step to secure a loan or should i find a home for and then go through the loan process?
I was planning on paying off our car and then use it as trade in for the new vehicle. Is taking on a new car loan right before buying a home a bad idea?
Typically, how negotiable are the listed prices?
For homes that are For Sale By Owner, would you still recommend a buyer’s agent?
What’s the typical timeline look like once we’ve found “The One”?

I welcome all your advice, even the unsolicited ones.