How can I attach a draft seal/excluder to a garage door

Your drill should definitely handle a 2 mm drill bit. Make sure it's a drill made for metal. Use a nail and hammer to countersink the hole. Then follow with the drill. I have always used a drop or two of oil, (3 in One works great), when drilling as it will cool and lubricate the surface as well as prolonging the life of your drill bit.

An alternative to drilling could be to glue the seal to the door. I think epoxy would do the trick, giving a strong bond even in cold weather. For example, Araldite. If you go ahead with this, use the slower setting ("Standard", I think?) glue - the Rapid stuff would set before you could apply and position the work. You will need plenty of clamps to hold it in place whilst the glue sets. However - I only mention this to give you another option - I think screws would be far better, as the seal could then be removed in future for maintenance / replacement, which might well be impossible with a strong epoxy bond.

I think you've probably got three things going on here:

1. Dull bit to start with. You need a sharp bit to drill steel by hand.
2. Inappropriate bit that dulls faster in steel. Often times, the drill bits that you'll buy in sets at the hardware store have a 118 degree angle on the tip rather than the 135 degree angle that is better for drilling in steel.
3. RPM not high enough. It takes a higher RPM to drill a small hole than it takes to drill a large hole. "Power" isn't the problem, it is speed. Power only comes into it as far as it takes power to maintain the speed under load.

I find that battery powered drills don't really have the speed it takes to drill with small bits. I like the German name for battery powered drills better - "Akkuschrauber." That's "battery powered screwdriver" rather than "drill." A screwdriver doesn't spin any where near as fast as a drill.

The holes you started look like ones that I made when I tried using a battery powered drill to make holes in steel. I gave up, and went and got my regular drill. That worked just fine.

1. Get a normal drill instead of the battery powered one.
2. Get a new bit (or sharpen the one you have.)
3. Use bits with a 135 degree point.

Feed rate enters into the problem as well.

"Feed rate" translates to "how hard you are pushing the drill into the metal" when drilling by hand.

Given where you need to put your holes, pushing hard is going to be difficult. You can't really "put your back into it" because the holes in an inconvenient position, and even if you could the metal will flex when you push hard. Along with that, drilling a 2mm hole and pushing hard on a hand drill is a good way to break bits - if you push just a little bit off of straight, you'll bend and break the bit.

Higher speed, sharp bit, proper bit, and a spot of oil are your best bet.

All drills have an optimum speed and feed rate. The smaller the drill bit, the faster it should be spinning and vice versa.

Looking at your two previous attempts I would say either not enough pressure on the drill bit (you may have to put something big and heavy behind the door if it is flexing when you apply pressure to the drill) or a dull blunt drill bit. Either of these will cause both the drill bit and and door to heat up and you end up "polishing" a depression into the metal instead of actually drilling.

When a drill is at its optimum speed and feed rate, chips of metal will come off more or less continously. Especially with hand tools, the pressure required to get the correct feed rate is often much more than you'd think!