New temp job: How do I talk to my manager about issues I am having in the workplace?

Recently got a 1-year temp assignment at a great company, with great pay. I was offered the job directly from the company rather than the temp agency and told that I would "likely have a full-time offer by the 6-month mark".

Great! Well, unfortunately, I am about a month in and I cannot get any quality work done due to a slew of ongoing problems. First off - I don't have a formal desk or work space. I've been placed "temporarily" in the chatty intern room with about 5 other people and more to come. Because of this, I am constantly mistaken for an intern - asked to do tasks unrelated to my team - and completely overlooked for important meetings and changing company knowledge. I fear that because I have been grouped in with the interns I am not being taken seriously and it is hurting my chances for full-time in the future. (also the interns do NOT stop talking - it's amazing! They can talk for 9 straight hours. How can one converse with another human for that long?)

I also do not have a quality working computer. I was given a (very used) company laptop that has had its hard-drive replaced more times than is healthy. It gets stuck on the welcome screen, randomly deletes files/drivers/applications, you name it - it's probably happening. Anyways - I've told my manager about my IT issues and because my work is centered around running heavy duty programs on my computer - it is making it impossible to do any work. He said he felt our company could do better but is now on vacation for 5 weeks. I might also mention that the interns have brand new MacBooks......

Bottom line: 🎶 I want to be the very best, like no one ever was, 🎶 but none of my equipment works and I do not have an adequate working space. I'm concerned this is negatively affecting my chance at full-time employment and do not know how to move forward in a corporate space. (Or stand up for myself without seeming like an ungrateful git)