What part of the world are you located in?
While pallets can often be found for free they can be a pain to work with even for folks with lots of tools.
Wood is still probably less expensive than making a cinder block expansion.
It's morning where I am.
If your mom is willing to physically help that is step one. Step two a budget. Step three is designing an easy way to expand/improve the coop within budget.
Big box stores like Home depot will make rip cuts for you. That takes a circular saw off the needs list. It also makes things easier to...
What about tools....like a drill or impact driver? Hammer and nails? While I recommend a good miter saw most young people would need adult help there... and with most power saws.
Are your folks even willing to discuss improvements?
Bird looks ok. I don't see any frost bite actually.
As to the housing issues.....
Have you looked at hoop coops?
They are much less costly than getting a shed and converting it.
The hard part is transporting the panels. 2panels make a structure 8'x8' and a bit over 6' tall.
Of course the...
Pics of the frost bitten birds please. We can help advise on treatment IF treatment is actually needed.
We are here and willing to walk you through improving their habitat.
What's the relative humidity been in your area?
Can you post pics of the most frost bitten comb?
Frostbite is very painful if messed with. It is not advised to even touch the damaged area. It's best to allow it to heal on its own while watching for infection.
If the whole comb is affected...
Vaseline in my experience is not helpful.
Are your birds actually getting frost bitten or are you trying to prevent it?
What are your overnight lows and what are your humidity levels reading?