Pot looks fine to me.
Peppers love the sun - so I would put it in a sunny window. (I live in Texas so I keep all mine outside, and plant most in the garden). If it is in a sunny location you might need to water it more often if it dries out. A common symptom of erratic watering is "blossom end rot" where you get spots of rot on the fruit - you don't have any, but something to watch for.
Generally the leaves look fine - dark rich colour is good. Maybe some concern for the leaves that are slightly curled - sun & regular watering should fix any problems there (and twice a week sounds fine if it isn't drying out).
Variety: There are about 4 species and many hundreds of cultivars. You may never know. The upward pointing nature of the fruits hints at something like a C.frutescens cultivar (eg. like many ornamentals) but I don't think this matches the leaves, and most C.frutescens fruit are small. The majority of cultivars are bred from C.annuum (including bells and jalapenos). Pepper shape looks like many jalapenos and there are varieties of jalapenos with different colours.
Try one. If it is hot, then I would go for "probably a variety of jalepeno".
You can eat a pepper whenever you want. There are different colour varieties but a typical pepper forms as green and is ripe when red - but you can eat it when it is green. The vast majority of green peppers in the grocers (including bells and jalapenos) aren't ripe! A ripe pepper will tend to be sweeter, but obviously people cook with unripe peppers for a slightly different taste.
As you have a few green and lots of orange, I think the orange are ripe. I would pick those soon.
Leaving too many peppers on a plant will inhibit further fruit formation. So I would pick them soon so you get more fruit, and before any pests come along and eat them for you.