What can do the conntrack-tools for me?

The daemon conntrackd covers the specific aspects of stateful Linux firewalls to enable high availability solutions and it can be used as statistics collector of the firewall use as well. The command line interface conntrack provides an interface to add, delete and update flow entries, list current active flows in plain text/XML, current IPv4 NAT'ed flows, reset counters atomically, flush the connection tracking table and monitor connection tracking events among many other.

So, does conntrackd provides an equivalent of OpenBSD's pfsync?

In short, yes. The daemon conntrackd synchronizes the states among several replica firewalls, so you can deploy failover setups with stateful Linux firewalls. However, conntrackd follows different principle designs with regards to OpenBSD's pfsync so they are not strictly equivalent.

Why use the command line tool conntrack instead of /proc/net/ip_conntrack?

There are several good reasons to do so. The /proc interface offers a quite limited interface to the Connection Tracking System since it only allows you to dump current active network flows. Instead, conntrack allows you to update network flows without adding a new iptables rule, e.g. update the conntrack mark, or dump the connection tracking table in XML format. Moreover, using the /proc interface to dump the connection tracking table under very busy firewalls, i.e. those with tons of connection states, harms performance. Specifically, this becomes a problem if you poll from the /proc interface to get firewall statistics. Also, conntrack offers connection events monitoring which a feature that the /proc interface does not provide.

Can I use conntrack to "cut" established TCP connections?

In short, yes. You can use conntrack to delete the state-entry that represents an established TCP connection without adding an iptables rule. With the appropriate rule-set, the packets of that TCP connections would be blocked and the client will experience a connection hang. Moreover, since conntrack is not dependent of the layer 4 protocol, you can use it to remove other non-TCP entries to obtain the similar described behaviour.