1. Paper should have novelty - i.e. the innovative component. Mathematics should be there to support your ideas, like proof, or showing the computational complexity of your proposed algorithm or other such things. Often times, the most reproducible ideas have simple but clever use of basics. If the novelty is in the mathematics of your approach, it should be there. It definitely isn't required, as such, novelty IS required. Moreover, a research paper is a research paper, it's the idea that matters and it doesn't matter what your status or position is in academia (BS, MS, PhD etc.). Also, ideally you aren't subjected to any favors for writing a paper at bachelors. Content matters the most.
2. Review paper may or may not have new ideas. However, review papers generally require a great deal of work and scientific maturity for proper and unbiased presentation of different techniques of that particular topic. Regular papers describe new ideas, new algorithms, new systems, improvements on the existing idea/algo/system/hardware/software etc.
US Patent website
"There are three types of patents. Utility patents may be granted to anyone who invents or discovers any new and useful process, machine, article of manufacture, or composition of matter, or any new and useful improvement thereof. Here is the process for obtaining a utility patent. Design patents may be granted to anyone who invents a new, original, and ornamental design for an article of manufacture. Plant patents may be granted to anyone who invents or discovers and asexually reproduces any distinct and new variety of plant."
Also, take a look at this:
Hope this makes it clear that not everything can be patented. Moreover, in Academia, A-class journal papers are looked upon more favorably than Patents.
Does this answer your questions?