Can you win big money joining your local chess tournaments and coming out in first place ?

While it varies based on the local chess clubs & public knowledge of the tournaments, you typically won't win "Big Money" by placing high (like Top 3) since the prize pool will typically be scaled to the number of participants in the tournament. You will likely have to pay dues to the local chess club (or a one-time non-member fee) in addition to the entry fee to participate in the tournament.

Participating at a state or national tournament (which likely requires membership in the national chess federation for your country to participate) would have larger prize pools. International tournaments may have the largest prize pools, but would require membership into FIDE (the international chess federation) & typically a high skill level (generally within the master range, but most likely in the grandmaster range to reach the Top 3 or Top 10).

No at the local level.

Only if you are secretly of Master or higher strength. And prize money is rarely 'big'.

Probably no

Yes