https://mfn24.myfootballnow.com/

Both teams 8-8
Both teams 5-7 in conference
no head to head
in same opponents games Philly 2-3 (W San Fran, Washington, L Minn, Carolina, Washington) and Chicago 1-4 (W San Fran, L Minn x2, Carolina, Washington)

Not sure what the process is.

Arizona had the same 8-8 record, so they have to be considered too. To me it looks like Arizona should have gotten in because they have a better conference record (7-5). Looks like there may be a glitch where if a team loses a tiebreaker for the division lead they aren't considered for the wildcard.

I did not even notice Arizona. You are right, they should be in.

What is normal way of determining who gets in?

http://www.nfl.com/standings/tiebreakingprocedures

Are you asking about Chicago & not Washington? Washington won your division by 2 games.

Regarding Arizona, Chicago, & Philly; Yes, it looks as though Arizona should have received the Wildcard bid.

Yes Chicago not Washington.

3 way tie, the tie breaker was probably common opponents.

According to the link I posted earlier from NFL.com, conference record is considered before common opponents.

JBD may be going to common opponents first