Is it safe/advisable to buy insurance online instead of through a financial advisor? If online, must I process the claims myself?

You can purchase certain insurances on your own. Insurances such as travel insurance are easy to claim and is more direct in their clauses in the sense that the claimable and non claimable criteria are all clearly laid out for x amount of premium you paid.

For other insurances like whole life, term, hospitalization, critical illnesses etc. Can be a little more complicated and differs across some insurance providers (triple claim, one time claim, early stage, late stage etc.) So for these it's usually more advisable to go to financial advisers who are more aware and trained in this aspect.