So if I'm correct, when you chose your starter gear and wealth you can either choose to roll some dice depending on your class and buy everything from that or you can choose the starter gear from your class and add the background items (including some gold). Here are my 2 questions:
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If I choose the gear from my class, do I have to take all the equipment from only one of the (a) or (b) choices, or can I freely mix-and-match choices from the two sets?
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Can you spend the extra gold you get from your background on extra equipment (e.g. some soap and some fishing equipment) or do you have to start with that gold in its pouch?
The reason I ask is that I have recently created a level 1 wood elf druid and I mixed-and-matched the gear from the class, so that I chose a shortbow as a "(b) simple weapon" for the first point and a "(a) Scimitar" at the second point. My DM wasn't so happy about this, as he can't understand why I'm mixing (a) and (b).
He also wasn't so happy with me spending my 10gp from my outlander background. I decided that my druid wouldn't lay so much worth in 10gp compared to a bunch of stuff I got for that (e.g. some soap and some fishing equipment and some other useful things) and he seems to think that I'm supposed to just start with the gold instead.
Best Answer
Class equipment
You did this right. You get all of the bulleted list of equipment, except where it asks you to choose either one option or the other using the (a), (b), (c) style of list. You don't have to "match up" the a's and b's, you can pick whatever from each line.
For the druid that means you get all three lines of equipment, which gives you armour, a pack, and a focus, and two choices to make: (shield or simple weapon) and (scimitar or simple melee weapon). Your bow is a legit choice for "simple weapon" from the former and the scimitar is also a legit choice from the latter.
The choices in each line aren't supposed to be linked, because if were, they would be directly linked together, not using a letter as a secret code. Instead of being written how they are, they would be written like this as a single bullet-item in the list:
(I'm not directly copying the last line because it's not necessary for the example.)
You can see this in action with the "Building Bruenor" example. Bruenor is a Fighter with the Folk Hero background, and on page 15 of the PHB, Bruenor's player chooses "a battleaxe and two handaxes". Those are two "mixed" options from the Fighter's choices on page 72: the two handaxes are a (b) choice from the third line, and the battleaxe is an (a) choice from the second line.
Notice also how the directly-linked way of writing things that I described above is used for many of the Figher's options on that page, so that (e.g.) taking a single martial weapon requires also taking a shield.
So you did this right. Gently correct your DM on this. Point to Bruenor's choices on page 15 and the Figher's choices on page 72, and how they aren't following an "all (a) or all (b)" rule. The alternative is simply too restrictive on creating PCs, and doesn't even make sense for other classes that don't have letters that all match up.
Starting gold
This is up to your DM, but they should consider being flexible about it.
The point of taking the starting package of equipment is to get started playing quickly. While the option of rolling for starting gold and buying equipment is meant to represent having acquired useful items over your lifetime rather than actually buying them all of a sudden with a sackful of hundreds of gold pieces, the equipment provided by backgrounds is supposed to be the actual gear you begin play with, including the "pouch of x gold".
So your DM is within his rights as DM to enforce that... however, the DM is not necessarily right to enforce it.
If your DM is trying to make sure you create the character "right", they may be blindly following the book where, actually, the book expects the DM to take some responsibility for running the game. The DM has authority to say how the game is run, but that also comes with the responsibility to say how the game is run. They can't disclaim all decision making, blindly, to the game books. That leads to making bad decisions when, actually, the book doesn't really care very much about something.
This is one of those cases where it's not actually important, and the DM should take responsibility for their decision.
If the player asks, "Hey, can I spend this gold on something? I don't think my druid would have kept a bag of gold around for all these years, it would have been traded for useful tools and provisions already," then it's entirely within the DM's rights to say, "yeah, sure, that sounds sensible." It's also within the DM's rights to say, "No, you should have that bag of gold, it'll make more sense for how the game is starting." The DM should never blindly follow the book when the alternative is not harmful to the game, especially when the point is so trivial as this.
So ask your DM to reconsider. Say, yes, the book does say you start with a pouch of gold coins, but ask if it will hurt the game at all for you to say you've already traded it for useful things instead. If the DM is adamant, shrug and move on — you can use it as an excuse for why your druid ends up in a town later: you want to buy a fishhook from these crafty but filthy city-dwellers!
If your DM is amenable to bending on this trivial point — and they might not be, if they're feeling insecure about their grasp of the game or their authority as DM — then it's fine. The important thing is to bring this to your DM for reconsideration, but to respect their authority over the campaign's operation.