Competition does matter, and the landscape does change based on different keywords. Most head terms don't have significant amounts of competition because they are too vague. Something like Wedding Cakes is not a good PPC term when you are looking to make money on your money as fast as you can.

It's always best to try bidding on the term (especially on Broad Match Modified) to see how it performs for you, but it is highly likely you will receive a low QS and a low CTR and higher CPCs which make it not as important of a term when you are looking at everything from a returns or profits perspective.

As far as low v high volumes for keywords, it's not something that I take into significant consideration unless my client is trying to get close to a specific number or is very concerned about a specific keyword. PPC is about throwing out a wide net for low, mid and high volume keywords, bidding on all of them, and learning over time what creates good return for your particular niche. What works for my business might be a horrible strategy for your business.