No one is going to ask you for your residence card - be more concerned about competing with 200 people on a single apartment that may be rented before you even get to your viewing.

I'll briefly explain, but there are other threads about this. There's also a ton of housing law that plays into this, which is far too complicated to explain here, but a distilled version of what you need:

Submit your residency application FIRST (but only after you've made plans and arrangements to go to NL - maybe even submit from within the country within a few days of arrival). You need this to start other things.

Rent anything. You are supposed to register where you live within five days. Hospitality stays may be exceptions. You would have to ask an Airbnb host if you went that route. Try to find anything local, even a sublet, and BE SURE YOU CAN REGISTER!!! There are a lot of illegal rentals, now, because of the market, and other people freaking out about their own registrations (plus just a ton of under-the-table stuff due to housing laws).

There are two types of registration. RNI and BRP. RNI is non-resident. You can do this immediately at any gemeente (and should, because it will give you your BSN more quickly, which you need for other things). As soon as you have an address you can register (you still need this ASAP), go to the gemeente with the lease, your passport, residence endorsement sticker, and the apostilled birth certificate, and you can be registered on the BRP (resident register).

Basically you need a legit address ASAP. Make an appointment with the gemeente ASAP (could be weeks of waiting, and if you're in a short-term, you will really be on a timeline to get registered).

Last thing not yet explained - the residence endorsement sticker is something you need from IND. You'll need an extra appointment for that. Make sure you have one full blank page (at least) in your passport. This is basically them approving your stay while the application is being processed. This is necessary for the gemeente to put you on the BRP. Without the sticker, they can only do RNI. You will need to get on the BRP.

The rental market is fucked right now. It will be a lot of extra stress to find a legal place, right now, long or short term. Just be prepared.

You absolutely don't need residency to get an apartment. Not sure where you heard that.

Generally, from what I understand (I'm in the middle of the process right now), you need a stamp from IND before you can be official on the DRP, so you can't even register your address until you're in NL completing the residency process. Don't worry about that.

What you need to worry about is the housing crisis. Certainly hope you've done enough research to know this is a thing. Beyond that, I can say it is BAD. I've been here about a month already, and have gotten ONE viewing (scheduled a couple others, but places were taken before I even got on the train). I've been looking anywhere within a couple hours by train from my temp place in Den Haag. I'm on two paid sites, and in about seven different apartment groups on FB. I've messaged countless listings (and that's the 10% that are good, as most are scams or illegal - the illegal ones are where they prohibit you from registering because they're doing something under the table).

I genuinely urge you to consider putting this off for a while. I am already having to consider what I will do if I can't find an apartment I can register soon. The paperwork is a trifle compared to finding somewhere legit to live and register. And you won't get ANYTHING without being here, on the ground, able to go to a viewing at the earliest possible time. I just scheduled one for Sunday (five days away), at the earliest time they offered, and already figure there's an 80% chance it's gone before my viewing slot (being generous, as I don't think they're showing it during the week). So, it's a gamble you'll have to decide for yourself if you'll take, but temp housing is limited and expensive, unless you go with one of those illegal sublets for a few (not saying you shouldn't, but you won't be able to register there).

So the most research you should be doing is on housing. I've heard the average time to find a place (even for locals with good jobs) is four months or so, give or take, depending on where you're looking. You'll have 90 days before your application needs to be fully submitted (and there may be some back and forth as you setup with KvK, and register with the gemeente, because parts of the process sort of pause as you get something from one place you need for another - you rotate a little between documents for KvK, IND, and the gemeente). Someone else recently posted a long plea for people not to come to NL for DAFT now based on his experience with the rental market. I had already planned the trip, found an affordable place to stay for a month, so figured why not give it a go, but really, think long and hard about whether you want to take the gamble now with the market in such a position. It will likely add a noticeable amount of stress to your experience getting settled (if you're able to get settled).

You should look in the netherlands sub, as well. There are a lot of other posts about DAFT, and the housing crisis, there.