Admittedly, luck does play a role. It's part of this (and most) games. And you can make whatever changes you want within your games, because that helps you have more fun. But I'll address both of your concerns anyway.
1) The game already has a mechanism for addressing first player advantage... sort of (as long as there are no attacks). In a 2-player game, there are 8 provinces (4/ player). With 3, there are still 4 each. With 4, 5, or 6, there are 3 each.
Let's say there is an ideal way to get provinces fastest. Everyone does it (and they all draw the same cards every turn). Pile runs out and everyone has the same number and it's a tie. But if someone doesn't play ideal (in a 2 or 6 player game or anything in between), and there's a chance someone could get more points. But let's say there still ends up being a tie. If the person who bought the last one had one more turn than the other, the game says that person loses.
Of course, it's still not that simple. There's a 3-pile ending possible. That may be fine/fair with 2 or 5 players. But with 3, that first player has an advantage to get one extra village... or laboratory, smithy, etc. And then attacks change things too... first player has a better chance of drawing their purchased attack first, giving an even bigger advantage. It's not perfect.
So some tournaments I've heard play multiple games between the same 2 players determine a winner of each round (for example, 6 games total with each starting 3 games). They can still go 3-3, but the chances of 4-2 or 3-2 and a tie (or others) definitely help break most ties.
Our own house rule is that whoever wins one game goes last the next... playing multiple games the same day or even if it's a week later. If I win, the player to my left starts the next game. It's okay if one person has a slight advantage because someone else likely will the next game.
You can still continue letting people play after the game ends until everyone had equal turns (likely giving people at least a chance to buy a duchy). It's a house rule, like putting money in the centre for Monopoly and collecting it when you land on Free Parking (that's not actually in the rules!)
2) Much shorter answer here. I believe also in some official tournaments, players have been able to choose their starting copper split. It makes sense since some boards strongly advantage one over the other.
It's even more powerful with some cards... Pooka (with Cursed Gold heirloom) and Fisherman are two I can think of. Imagine drawing 5-then-2 with Fisherman against someone else who draws 2-then-5.
I like the randomness in the starting draw. It adds an additional challenge when starting with a non-ideal opening. Although, now I'm thinking we should probably let my daughter (6 years old and loves playing, competitively too!) the chance to choose her opening to balance it out a bit.