Will strikes and delays affect my ESTA application?

If airlines go on strike and your flight is cancelled, this shouldn’t affect your ESTA application. Once your ESTA is approved it remains valid for 2 years or until your passport expires, this depends on which one comes first. However, if your planned travel date is in the last days before your ESTA expires and your flight gets cancelled, you will need to apply for your ESTA again, if your current one will not cover your rescheduled flight. Even though this would be a matter beyond your control, you are required to apply for your ESTA again and pay the fee to submit a new ESTA application.
In the case of a government shutdown in the U.S. due to any budgeting reasons or other reasons, this should not affect your ESTA application. Any essential departments still remain operational during a government shutdown. The department of Customs and Border Protection is viewed to be an essential part of the government so it would remain largely unaffected by a potential government shutdown. Also, because it is fee based, this shouldn’t slow down your ESTA application as fee based systems should not be affected during a shutdown.
If you have gone to the U.S. under ESTA and some form of emergency prevents you from departing on time, the U.S. Citizenship and Immigration Service (USCIS) district director (who has jurisdiction over the place you are temporarily staying) may, as they fit, grant you a period of within which you can leave. This must not exceed 30 days and if you depart in that period of time, you will not be judged to have overstayed your permitted time.
This is only granted for real emergencies, such as hospitalisation, conditions that cause flights to be cancelled or delayed for more than 24 hours (i.e. weather, worker strikes, etc. etc.). Otherwise, people visiting under the VWP may not extend their period of stay beyond their initial 90 days. If necessary, you may apply for a visa during this period if required. Do not confuse this with reapplying for ESTA. If you apply for ESTA again, your renewed ESTA will still not allow you to stay beyond the 90 days from the previous ESTA authorisation.