Others have pointed out most of the major things that you should know. The only one which I would add is that if you don't want a piece of information known, you should not input it in any of the data fields on your profile. Although the privacy controls make it appear that your information will not be disclosed, your information could still leak via 2 direct routes, and one indirect route:
2. It is possible to determine names of members who hide their names as their profiles come up as matches when searching or checking for duplicates. So you could have Robert Anderson who has a privacy setting that only shows "B Anderson", however when searching for "Robert Anderson" matches, "B Anderson" will show up in the list of matches.
3. Provided that you connect your profile to your ancestors, you're profile will be connected to their profiles even if you set your profile to private, as entries of your user-ID will show up connected to edits of those ancestors. If you can be connected to your family tree, other people will likely be able to determine some aspects of your identity, e.g. your surname, your age, in what region you live... and perhaps even specifically who you are. This is especially problematic for Americans due to lax privacy laws.
The first issue is likely just a matter of "when", and it's a risk that exists with any and every website. The second issue is avoidable, but it is how WikiTree currently functions. (And WikiTree currently fails to explain and disclose this in the Help pages, as would be necessary for informed consent and user trust.) The third issue is dependent on your choices, and the both the knowledge and skill of your adversaries. Theoretically one would need two accounts and some very careful two avoid this entirely.
Your real name is preferred for your account profile, especially your surname (family name), but you can use "Anonymous" or initials.
As others have mentioned, the use of "Anonymous" as a surname is explicitly allowed. There are 1130 profiles with the surname "Anonymous" on WikiTree. Likewise there are 18906 profiles with the surname Murphy (oh, joy!), 84181 Johnsons, 190008 Smiths... and zero McLovin family members. Vires In Numeris, eh?
Also, as a warning: Sometimes WikiTree doesn't behave as users expect. There are other sources of data leakage. Otherwise private email addresses disclosed when you merge things; Home addresses revealed through uploaded photos. So if you wish to be private, be cautious.
Ultimately, if you wish to correspond with others or reveal your connection, your real identity risks leaking. So in my view, it is not feasible to "use [a] real name only for [...] account/private messages but use a pseudonym for anything public?" WikiTree doesn't work that way, since practically everything (besides living relatives) is public.