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 so for 2 years or until your passport expires, this depends on which one comes first. Unless you are travelling during one of the last days where your permit is approved and your flight happens to get cancelled. Even though this would be a matter beyond your control, it would be best 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 do not take a hit during a shutdown.
If you have gone to the U.S. under ESTA and some kind 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 not exceeding 30 days. If you depart in that period of time, you will be considered to have satisfactorily accomplished the visit and not to have overstayed. 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 need be. 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 our 90 from the previous ESTA authorisation.