Is this normal for the spoke nipples to stick out this much ? Even after installing rim tape I can literally feel them, wouldn't it press/pinch into the tube when riding a bike ?
Yes. You have a single-walled rim (which is a terrible name because it's really a single "valley floor") These are sometimes called "box section" rims because they're approximately a rectangle with a side removed.
This is the way bike rims were made for decades. Notice the shoulders of the nipples are rounded off, and nowhere does a spoke poke above the top of the nipple.
You've peeled back the rim tape, which appears to be a fairly thick layer of butyl rubber, same as your tube is made from.
The other style of rim is called double-wall and the nipple is completely below the visible floor.
Road bikes started moving toward double wall rims as rim depth increased. This coincides with the awareness of aero benefits.
On the other hand, MTB riders have valued the additional strength of double-walled construction for a long time - they're quite hard on rims so tough ones are prized.
The downsides of double walled are that they're hard to make from steel, so the rise of Aluminium as a rim material coincides with popularity. Secondly, there is more material at the outer edge of the rim, which makes acceleration feel slower.
Back to your question - its okay for the nipple heads to indent on the tube. That helps stop it sliding around inside the tyre and giving you crooked-valvestem.
There should be no sharp edges to abrade or cut your tube. Likewise the spoke should not extend through the nipple at all. You can use a file or dremel-style tool to knock down any sharp spots.
Another option is to fit a second rim strip, or a better one. There are some made from a nylon-like material that has just-enough stretch to get on the rim but shrinks down again and protects the tube. Or you can bodge it with layers of paper masking tape on top of your existing rim strip. Just be sure to get a width that sits down in the valley and does not interfere with the tyre/rim bead hook.