My family and I will be flying QF MEL-LAX (QF93) and the ticket is Classic Rewards. Separately I have booked WestJet tickets from LAX-YVR 3 hours after scheduled LAX arrival.
I should warn you that you are taking this trip at your own peril. If your Qantas flight arrives into LA late or immigration takes longer than expected and you miss your LAX - YVR flight, you will be on the hook for finding new flights out of pocket since you did not appear for your YVR flight. This is one of the many reasons why I encourage people to book flights as one single itinerary (i.e. MEL to YVR via LAX versus MEL to LAX and LAX to YVR as separate tickets). That way, if Qantas gets you into LA late, they are on the hook for finding that new flight, providing you with a hotel, meals, and even provide cash compensation of up to $1000 CAD, as required under Canada's APPR legislation.
Whether 3 hours is enough time on an international itinerary to make the connection is really a judgement call, I suppose. I have had 4 hours connections on domestic flights where I missed the connecting flight, but have also made 50 minutes connections on international flights where I comfortably made it.
I have read that WestJet is called "Qantas Canada" so I'm wondering if Qantas would be able to link my QF and WS bookings so that bags will go through all the way and I don't have to lug them from Tom Bradley to T2 in LAX. Any thoughts on that?
Qantas Canada is my (humorous) attempt at naming WestJet as their motto is identical to that of Qantas' (i.e. "The spirit of Canada"). However, they are in fact two separate airlines which have a weak partnership with each other. To your point about tagging bags all the way to Canada, well it doesn't matter, really. When you arrive at LAX you must clear US immigration, you must collect your bags and you must clear US customs with it. Once you are through customs you would then need to re-check your bag. You might see a WestJet transfer desk there to accept your luggage and retag it, but if not, you'll need to make your way to the WestJet terminal, and re-check your bag that way. Remember to get rid of the old bag tag (and any barcode sticker attached to the bag) so that WestJet has a "clean" bag to tag, lest you want your bag directed to the wrong airport.
When you enter the US, there are ways to speed up the whole immigration process. Specifically if you are a US, Canadian or a
national of a Visa Waiver Program country, including Australia, you can now
use the Mobile Passport Control, which allows you to complete your customs and immigration declaration before reaching immigration. From there you enter a queue marked Mobile Passport Control and you will move through very quick (often there is no queue there). Obviously if you've got Global Entry (or Nexus) use the Global Entry lane as that's even quicker and doesn't require you filling out a declaration or showing a passport.
Also curious if I will get any benefit in WestJet from either QF Platinum or Skyteam Elite Plus?
As a Qantas elite expect no added benefits of plugging in your QFF number (aside from earning a marginal amount of points).
WestJet does provide additional benefits to holders of certain SkyTeam status. For instance,
with Delta Airways, you can select the best seats in economy and have additional baggage allowance as a Gold medallion or better. You might also be able to access the Delta SkyClub lounge but the details there seem a bit murky (Delta famously now requires travellers with status flying Delta internationally to be seated in Premium Economy or better to get lounge access through status). If your status is with Air France/KLM, I cannot vouch for what the benefits are as they don't seem to be listed (other than earning miles and potentially XP points on revenue tickets). Your best bet would be to try plugging in your Skyteam Elite Plus frequent flyer number into your WestJet booking and see what happens. If you can now start selecting seats for free and your baggage allowance updates to indicate free bags, then chances are that is the frequent flyer program to use.