What's the cheapest disk type to use?
It depend since if you are using managed disk then you get total price and if you use un-managed disk then you might pay for the transactions, so it is not a black and white answer without more information. In first glance seems like the cheapest disk cost $0.33 per month and it has 4 GiB (it is on preview option).
You can examine the prices in the following two pages:
https://azure.microsoft.com/en-us/pricing/details/managed-disks/
https://azure.microsoft.com/en-us/pricing/details/storage/page-blobs/
Is there a way to set the size of a disk to only the size required for the image I choose?
Not even in theory since the s always need more than it actually use for the installation, unless you are using specifical version of OS (not related to the Azure) which meant for example to be running directly from disk drive.
If I decide to destroy my existing VM, is there any penalty? Can I use the same names? Can I use the same DNS subdomain (<subdomain>.<region>.cloudapp.microsoft...), or do I have to pick another one? Is it as simple as just deleting it, or are there other things left over?
Why not try it?!?
Sometimes the best option is simply test
Note! I am note sure what you mean by "destroy my existing VM". I will answer about deleting the virtual machine, hoping this fit your needs. If you use tool like Terraform then please specify it.
Can I use the same names?
Yes
Can I use the same DNS subdomain
Not immediately. You might need to wait
Are there any other things I can do, besides choosing the smallest VM size (Standard B1ls), using the cheapest disk type and smallest disk size, and only keeping the VM on when I'm using it (which is maybe an hour a week), to reduce the cost even more?
There are multiple way to reduce the cost and not all can implement together. You will need to check all combination but I can give you one more new optional feature to save money if it fits your needs: Azure Virtual Machine SPOT
The SPOT option allow Microsoft to close your machine if they need the resources but it cost less. Read more here: https://azure.microsoft.com/en-us/pricing/spot/
B1 is not the smallest. E1 is a Preview price of 0.15 for a month. I think this is the cheapest