Is it allowed to have my name tape and "US Army" tape sewn on without having my rank sewing on?

No, all three must be sewn on, or velcro on, no in between. Advice, I waited until I was a PFC, I then sewn on the tapes and rank of one top, but waited until I was a specialist to sew the other three uniforms I had. It isn't much of an expense, but if I were you, I would wait, time will fly by and you will be a specialist before you know it.

PV2 (Join to see) What has your leadership told you when you asked this question?

All or nothing private. I would have one uniform with just Velcro so when its time for promotion you can use that uniform. Now badges are a different story when it comes to sew or Velcro.