Welcome to the wonderful world of planted aquaria.
There is good news and bad news. The good news is that the algae problems are normal and can be dealt with. The bad news is that some of the plants are not true aquatic plants but terrestrial, and will not last. It is frustrating that some stores still sell these for aquaria.
Taking the last point first. The dark green grass-like plant in the rear corners is Ophiopogon japonicus, commonly called Fountain Plant or Mondo Grass. This plant can last for a year or perhaps longer fully submerged, or it may not. It looks fine so far, but if it begins to rot apart, pull it out.
The plant in the centre at the back that has wider leaves up to the surface is another terrestrial, a species of Dracaena, the same genus that contains common houseplants. I would take this out, as it is not likely to last and it can create a mess as it rots.
The other plants are aquatic. Those attached to the wood, and one is in the gravel, are Anubias. The name shaded which means you can click on it for our profile of this species that will give you more info. One important point is that the rhizome should not be buried in the gravel or it can rot. This plant attaches well to wood and rock. If you lift the rhizome out of the gravel and leave the fine roots buried, it should be OK. The plants on the wood are fine [I'll come to the "problems" momentarily].
Photo 7 shows (I think) dwarf baby tears, but if someone else has a different opinion I won't argue as I do not know these tiny plant species well. Not usually easy to grow, as it requires more light and nutrients. It also will do better in a smaller-grain substrate like sand.
To the last photo, of the two small plants; the one on the right is I think a crypt, the left looks like either a Dwarf Sagittaria or possible a dwarf form of the Mondo Grass.
Now to the problems. New plants take acclimating to a new environment, so yellowing of existing leaves which will die off is to be expected. However, plants need nutrients, so of which occur in the tap water (the "hard" minerals like calcium, magnesium, etc) and some occur in fish foods. But in new tanks especially nutrients will be minimal, so...are you adding any plant fertilizer, and if yes, which?
Light is the single most important issue. Can you detail your tank lighting? Be specific.
To the algae and related stuff. Diatoms (brown algae) is common in new tanks, so that is not a concern, but wipe it off the plant leaves during the water change. The greenish could be hair algae or fuzz algae. This is due to an imbalance between light and nutrients, so you data on the light and ferts will help us sort this out.
I would lower your tank temperature, 80F is at the warm end for the Pristella Tetra, and the Betta can manage a tad down too. Around 77-78F. This combo may have issues, combining tetras with male Betta is not the best. The tetras may decide to fin nip. And as they grow, the 10g will be cramped space. Check our profile.