I also made the same mistake! After thinking about it, I think the answer we might be looking for comes in two parts. Firstly, because we are using the future tense, we need the perfective form, заключить rather than заключать (which would incidentally conjugate они заключают). Secondly, if заключить conjugated like говорить, we might expect to have они заключят. However, this would be wrong, as there is a spelling rule in Russian that after some consonants (including ч), я is replaced with а. So finally the correct conjugation here is in fact они заключат!
Заключать and заключить are perfectly regular. They conjugate like читать and говорить respectively. These verbs do not even change stress (though, in speech заключить and включить are experiencing the same shift that варить already went through: many speakers move the stress to the stem in all non-past forms except 1st person singular).
In this context, yes.
До́гово́р is a bit broader term than контра́кт: до́гово́р can sometimes include oral and informal agreements, not just official contracts. However, in most cases it means the same thing as контра́кт.
Подписа́ть means 'to sign'. It doesn't always refer to making a contract: when one side подписа́ла до́гово́р, it's not enough. All the sides need to sign it and only then they will заклю́ча́т до́гово́р. Sometimes, it refers to writing signatures in other contexts, e.g. «подписать книгу» can mean giving an authograph.
N.B. When I mark two stresses, it means 'choose one.' Они́ заключа́т догово́р is the stress used in formal speech, они́ заклю́чат до́говор is a colloquial variant.
We can use the simple present in cases like this. http://www.ef.com/english-resources/english-grammar/simple-present-future-events/ Tomorrow they sign the agreement.
Yes, but Russian also allows using the present tense to talk about future events. So you’re expected to translate «За́втра они́ заключа́ют до́гово́р» as 'Tomorrow they sign the agreement', and «За́втра они заклю́ча́т до́гово́р» as 'Tomorrow they will sign the agreement'.
N.B. Two stress marks mean that either stress is possible. Заключа́т догово́р is the formal pronunciation, заклю́чат до́говор is colloquial.
Arguably yes - but because that's present-future and the Russian is true future (I know you know that), maybe they didn't want to accept that. Doesn't your sentence correspond to "Завтра они заключают договор"?
Or maybe it's just an oversight. Did you report it?