I'll base my answer on the American depositary receipt (ADRs) AXAHY and TOT as well as their underlying ordinary shares on its local exchange (Euronext), which I own in US-based (for ADRs) and France-based (for ordinary shares) brokerage accounts.
Are ADRs more expensive? (e.g. custodian fees)
Yes. E.g., no yearly fee on AXA, but 0.45 USD/share yearly fee on AXAHY. See https://money.stackexchange.com/a/128258/5656 for the fee breakdown.
From https://www.sec.gov/investor/alerts/adr-bulletin.pdf (mirror):
"Depositary banks may charge other fees, such as
relating to the distribution of dividends, foreign
currency exchange, voting of shares, and other matters."
Are ADR dividends delayed?
Yes. In 2020, I received the AXA dividends on 2020-07-09, whereas I received the AXAHY dividends on 2020-07-24.
Will I be able to receive corporate communications (e.g. annual reports) if I buy the ADR?
I don't know if ADR owners receive all corporate communications, but they do receive at least some of them. For example, I received an optional dividend (mirror) event a few weeks ago for my TOT ADR shares owned in a US-based brokerage account.