Learn Russian: В рамках реализации проекта мы наймём новых специалистов. - As part of the project's implementation, we will hire new specialists.

The compound preposition в рамках (within the framework of / as part of) requires the Genitive case, which gives us реализации. Then, проекта is also in the Genitive because it shows possession or relation (the implementation of the project). This creates a Genitive chain, which is very common in formal Russian.

Since this lesson also covers в ходе, could we say в ходе реализации проекта instead?

Yes, but it slightly changes the nuance. В рамках implies within the scope or as a structural part of the project. В ходе means during the course of, emphasizing the timeline and process. Since hiring is a planned part of the project's structure, в рамках fits perfectly, but в ходе works if you mean while we are implementing it.

What is the infinitive of наймём, and why does it look so different?

The infinitive is нанять (to hire). Verbs ending in -нять (like понять, занять, принять) insert an -м- or -н- into their stem when conjugated. So, нанять becomes я найму, ты наймёшь, and мы наймём. Because it is a perfective verb, these forms create the future tense.

Why does the ending of look like the Genitive case if it is the direct object of the verb?