Always start with the verbs. As they determine the phrase in question.
Stimmt es, dass Sie … ? (interessieren – Umweltschutz)
Your first example has the incomplete clause starting with dass, so this must be a dependent clause with a conjugated verb at the end.
Stimmt es, dass Sie … ‹Umweltschutz› … ‹interessieren›?
Apart from a very few exceptions, the first position in a dependent clause is the subject. So Sie is the subject and this is third person plural. So you have to conjugate for third person plural.
Stimmt es, dass Sie … ‹Umweltschutz› … interessieren?
Next, think about that verb phrase. It's sich für etwas(Akk) interessieren. That's leads you to
Stimmt es, dass Sie sich für (den) Umweltschutz interessieren?
You are done. Both the variant with den and without are valid. The latter treats Umweltschutz as a concept, which is likely what you want.
Ich würde mich freuen, … . (dürfen, neu, Projekt, vorstellen)
Here you have two verbs, dürfen and vorstellen. Dürfen is a modal. They take an infinitive. So you already know vorstellen must be an infinitive.
There is no conjunction hint to tell the type of the incomplete clause. But the example sentence starts with a main clause. You can tell that from the conjugated form würde in second position. So let's assume the second clause has dependent clause word order.
Ich würde mich freuen, … ‹neu – Projekt› … vorstellen … ‹dürfen›.
The verb phrase is etwas(Akk) vorstellen so you need an accusative object. That means Projekt can't be the subject. That means you can choose a subject for this clause on your own. How about ich? Conjugate dürfen accordingly:
Ich würde mich freuen, … ich ‹neu – Projekt› vorstellen darf.
Which conjunction fits? Dass doesn't because the main clause features Konjunktiv II: it's not a fact, it's a non-fact. A wish. Matching conjunctions are wenn and falls.
Ich würde mich freuen, wenn ich ‹neu – Projekt› vorstellen darf.
You are almost there. Decline the noun phrase.
Ich würde mich freuen, wenn ich ein neues Projekt vorstellen darf.
This is okay. But you can conjugate dürfen a bit better.
Ich würde mich freuen, wenn ich ein neues Projekt vorstellen dürfte.
This is the subjective use of Konjunktiv II with dürfen. It's similar to saying ich möchte instead of ich mag. It tones down your demand from fact to non-fact level.
You are done. However, there's a variant.
As you had chosen the subject ich which is identical to the subject ich of the main clause, the agency of the dependent clause is the same as that of the main clause. That makes it possible to use an infinitive clause instead. The subject and the conjunction are implied in that variant, and we need the zu marker instead:
Ich würde mich freuen, ‹neu – Projekt› vorstellen zu dürfen.
You are again almost there. Decline the noun phrase.
Ich würde mich freuen, ein neues Projekt vorstellen zu dürfen.
Done.