I get asked for a gold shovel every day, but have never seen one.
Aside from the availability question, do you really want to give a peep a gold shovel? They cost a bunch of coins, and the peep is going to give you a lemon maranta seed in return, or perhaps a brass axe. Yawn. The peeps have a happiness meter, but dissing them for one request doesn't seem too bad.
I have a brass watering can that never wears out. It is the same one I got on my first day.
IMO, a good way to save coins is to use the brass watering can and brass bug net until you're on a more firm financial footing. I usually wait for the gold bug net, skipping the silver, and buy the silver watering can when I feel comfortable spending 300 coins (more or less) that I don't really need to spend to complete my objectives. It makes an essential chore less tedious. I haven't seen a gold watering can.
I stopped cutting trees after I thought I was not getting any benefit from them. I thought I would need the wood to build with, but have not had to except for my own house. Thinking I will need to chop trees as I get deeper into the game though.
I think that clearing the forests, or at least thinning them out considerably, makes the game more fun to play because you can see treasure pits and stuff that are concealed by the trees. Later, you can replant the forest to your satisfaction and landscape the game world to your tastes. That's a lot of fun to do!