1. Yes, this can be MTU problem.
2. There is no such thing as "correct MTU". ASA/FTD firewalls support Path MTU Discovery (PMTUD) both between the sender and the firewall and between firewalls terminating IPSec tunnel. PMTUD relies on "ICMP unreachable fragmentation needed and DF set" messages. If they are blocked somewhere in the path, PMTUD fails and big packets can be lost. In this case you can clear DF bit on the firewall and fragment traffic.
3. The "crypto map <name> <priority> set df-bit clear" is sometimes used if PMTUD doesn't function, but fragmentation should typically be avoided as performance can drop a little bit. If traffic volume isn't huge, CPU shouldn't increase much on ASA/FTD fragmenting packets. If traffic is fragmented by ASA/FTD, the reassembly is performed by the receiving endpoint (e.g. by the server). If IPSec traffic is fragmented somewhere on the Internet, the reassembly is performed by the receiving ASA/FTD. This is a worst-case scenario, watch out reassembly drops ("show fragment"). The only other solution is to lower MTU on endpoints.
4. No, MTU discovery is supported by AnyConnect/ASA(FTD), but not in case of L2L.
