Is it normal for my header pins to look like that after soldering?

If it boots and the LED comes on, the joints are probably making contact right now, so you don't need to panic. The issue with chunky solder and little bridges is that they can work fine for weeks and then develop intermittent faults later, usually from a joint that looked connected but never actually wetted the pin properly. Those are the ones that bite you six months in when something randomly drops off the network and you spend an evening chasing a ghost. The easiest check is a visual one under good light or a phone camera zoomed in. A good joint looks like a small shiny cone where the solder flows up the pin and spreads onto the pad. A suspect one looks like a ball sitting on top, dull and lumpy, with a visible line where the solder meets the pin. If you see any of those ball-shaped ones, touch your iron to them with a bit of fresh solder for a couple of seconds and they'll usually sort themselves out. The bridges are a separate matter. A tiny bridge between two adjacent header pins on a Pico is often harmless if those pins happen to be connected anyway, but you can't rely on that. Get some solder wick or a cheap desoldering pump, heat the bridge, and pull the excess off. It takes thirty seconds per bridge and saves you a lot of head scratching later.

For next time, flux is honestly the difference between fighting the joint and the solder just flowing where you want it. A small tin of flux paste costs next to nothing. I also found it helped to tin the pad and the pin separately first, then bring them together with the iron, instead of trying to do it all in one go. Since you're prototyping on a breadboard setup anyway, I'd just fix the bridges and reflow anything that looks dull, then run it for a while. If nothing acts up after a week of use, you're basically fine and you'll laugh at these joints when you look back at them.

I'm skeptical about just heating up a dull joint and calling it fixed. While I get that the advice suggests running it for a while to see if it acts up, my experience says that just reflowing a cold solder joint often doesn't fix the underlying issue; it just applies heat to a poor connection. If the joint never properly wetted the metal, you're just forcing a poor connection to hold, and that just shifts the point of failure somewhere else down the line. You usually need to completely scrape off the bad material and start fresh to get a solid base.

My first Pico header job looked a lot like yours and I chased the lumpy joints for ages before realizing the problem was my iron. It was one of those unregulated plug-in things that heats up whenever it feels like it. Picking up a cheap temprature-controlled iron was the single thing that fixed my soldering overnight. Set it around 320, and suddenly the solder flows instead of sitting there half-melted while you panic. Lead-free solder makes it worse too, it's sluggish compared to the old stuff. If your joints are chunky, chances are decent the iron just isn't keeping up, and no amount of technique fixes a cold tip.

I totally agree that getting those tiny bridges cleared out is key if you want reliability. It's frustrating when something looks fine but then fails later on. I found that even when using a temperature-controlled iron, if you don't use decent flux, the solder just refuses to properly wet the metal. I once started with cheap solder and pad material and all my joints looked dull and chunky, but switching to a little batch of liquid flux made a huge difference in how quickly and smoothly the solder flowed around the pins.

Quick update - I went with the advice about using better flux. I figured out that my solder just refused to wet the metal properly on the lumpy spots. After applying some flux and carefully cleaning the bridges, the joints look really neat now. The board is holding steady and everything is reliable!