I will answer your questions last in first out, because your last questions are easier to answer
- If I decide to buy a new construction, should I get an inspector?
Yes, without question. You are spending 500k. An inspector is only $500. Even on a brand new house, any inspector (not employer by the builder) will find more than $500 worth of things you can get the builder to fix, even on a brand new house.
- Should I get an agent to buy a new/first home? What are the benefits? Who’s going to pay for the agent?
The buyer's agent (who is actually referred to as a "selling agent" in the contracts, I believe) is paid with half the typical 6% commission. If you don't have a buyer's agent, then the seller's agent (aka "listing agent") keeps the whole 6%. So you don't really gain anything by not using one.
However, that does NOT mean the agent is on your side. They have an incentive to get you to buy a house at the highest price possible (larger commission) and/or to buy any house at all (if you never buy a house, they never get paid).
In Seattle you have a couple of interesting options for your buyer's agent. Both will give you a portion of the 3% commission back. One is Redfin, that last I checked gave up to half the commission back to the buyer. And another is WaLawRealty, that charges a flat fee and gives the buyer the rest; however, if you never buy a house, you have to pay them out of your own pocket.
- I looked at some new constructions and I’d like to learn more about the reputation of some builders...
I can't answer this question. (Just mentioning this for completeness)
- Who could help me evaluate if buying a home is more profitable than renting? Are there professional who can help with that? I know that there are calculators on the web, but I would like something specific to my situation. Right now, my rent is about $1375 + Utilities. A house would be around $500,000.
Only you and your family can decide that. A case could be made for either renting or buying. Interest rates and lowered (from the peak) prices definitely help.
But the true decision point is: Is now a good time in your life to buy a house? When you buy you will be "settled down" to one spot, for better or worse; have you lived in the area to be ready to commit to it long term? Is your family about the size it will be for the next decade or two? How much of a down payment do you have saved up? How is your credit history and job stability? Do you have kids; have you looked into the schools in the area you're targeting?