Yep, you can absolutely book two different cabins in each direction roundtrip with miles as long as they are not BE awards (which can't be combined with non-BE awards, just like BE revenue fares can't be combined with other revenue fares in a different cabin). You just have to make you specify Main to start your search (and not a higher cabin -- otherwise, it will just show that cabin and higher as options).

Whether or not there are roundtrip awards depends on the route. For domestic routes, many only have one-way awards as long as it's fairly competitive with LCC service. LCCs and Alaska only have one-way fares and DL tends to match them (with both one-way revenue fares and awards). For US-EU routes, most routes will have roundtrip awards as the cheapest options and booking as two one-ways will generally cost more. One-way awards can also incur carrier surcharges when booking on partner metal. Roundtrip awards on main partners from US to EU will not have carrier surcharges (although they won't qualify for Amex 15% discount).

Note that roundtrip awards also almost always have minimum stay requirements. So if you are staying for shorter durations (and don't at least have a Saturday night stay from point of arrival or a 7 day stay), you may likely end up getting one-way awards in both directions and not the cheaper roundtrip awards.

For example, the cheapest roundtrip Main cabin awards on ATL-EDI valid for August travel are 66.2K roundtrip with Amex discount and require a 7 day minimum stay and mid-week travel. The cheapest one-way Main awards on ATL-EDI valid for August are 45.9K with Amex discount. There are also roundtrip awards for higher cabins of service. You can still qualify for the cheaper roundtrip awards with different cabins in each direction as long as you meet the minimum stay requirements for them. You can combine Main and C+ awards roundtrip on ATL-EDI from 75.6K total (42.5K + 33.1K). Booking as two one-ways would require substantially more miles.