Would I be able to use the same shields that I am using with the Uno?
Most likely.
Can I use the same sketches?
Some changes might be needed. For instance, you don't need to use SoftwareSerial on the Mega.
- Keep improving my sketch but I wonder how much dynamic memory should I free for my sketch to be stable. Do I need to reduce the use of dynamic memory down to 85%? 70%?
Keep reducing memory usage until the message about instability goes away.
Post your code. There may be things you are overlooking.