Not being able to login to the admin console, I checked the httpd log and found the following errors:

[Wed Jun 07 12:50:59.352022 2017] [:error] [pid 10240] Unable to verify certificate 'Server-Cert'. Add "NSSEnforceValidCerts off" to nss.conf so the server can start until the problem can be resolved.
[Wed Jun 07 12:50:59.353372 2017] [:error] [pid 10237] SSL Library Error: -8181 Certificate has expired
[Wed Jun 07 12:50:59.353395 2017] [:error] [pid 10237] Unable to verify certificate 'Server-Cert'. Add "NSSEnforceValidCerts off" to nss.conf so the server can start until the problem can be resolved.
[Wed Jun 07 12:50:59.986025 2017] [core:error] [pid 11522] AH00546: no record of generation 47 of exiting child 10203

I also get an error during enrollment of a new client (which seems to retrieve a valid certificate anyway):

Password for admin@HQ.SPINQUE.COM
Successfully retrieved CA cert
    Subject:     CN=Certificate Authority,O=HQ.SPINQUE.COM
    Issuer:      CN=Certificate Authority,O=HQ.SPINQUE.COM
    Valid From:  Mon Mar 16 18:44:35 2015 UTC
    Valid Until: Fri Mar 16 18:44:35 2035 UTC

Joining realm failed: libcurl failed to execute the HTTP POST transaction, explaining:  TCP connection reset by peer

Services are up:

$ ipactl status
Directory Service: RUNNING
krb5kdc Service: RUNNING
kadmin Service: RUNNING
named Service: RUNNING
ipa_memcached Service: RUNNING
httpd Service: RUNNING
pki-tomcatd Service: RUNNING
ipa-otpd Service: RUNNING
ipa-dnskeysyncd Service: RUNNING
ipa: INFO: The ipactl command was successful


Certificate monitoring seems ok:

$ getcert list -d /etc/httpd/alias -n ipaCert
Number of certificates and requests being tracked: 8.
Request ID '20160501114633':
status: MONITORING
stuck: no
key pair storage: type=NSSDB,location='/etc/httpd/alias',nickname='ipaCert',token='NSS Certificate DB',pinfile='/etc/httpd/alias/pwdfile.txt'
certificate: type=NSSDB,location='/etc/httpd/alias',nickname='ipaCert',token='NSS Certificate DB'
CA: dogtag-ipa-ca-renew-agent
issuer: CN=Certificate Authority,O=HQ.SPINQUE.COM
subject: CN=IPA RA,O=HQ.SPINQUE.COM
expires: 2019-01-26 19:41:51 UTC
key usage: digitalSignature,nonRepudiation,keyEncipherment,dataEncipherment
eku: id-kp-serverAuth,id-kp-clientAuth
pre-save command: /usr/lib64/ipa/certmonger/renew_ra_cert_pre
post-save command: /usr/lib64/ipa/certmonger/renew_ra_cert
track: yes
auto-renew: yes

Version:

$ ipa --version
VERSION: 4.4.3, API_VERSION: 2.215

Could you please point me at what else to check?