They have the same meaning, but Russian distinguishes between singular/informal and plural/formal in the 2nd person (for example with "ты" and "вы"). With 2nd person imperatives (which is what "извини" is), the plural/formal marker is the "-те" at the end. So, you'd use "извини" with someone you know in an informal setting (or with a child), and "извините" with unfamiliar persons.
The suggested correct translation on the problem I encountered was Сожалеем! Which I looked up to mean sorry in the meaning of "pity", I think... maybe. I was under the impression that the usage is different than Извини! Could someone clarify the differences between these words. Also where does Мне жаль and Прости! fit into this "Sorry" spectrum.
You are correct about your alternate lookup Сожалеем. The more common use of that form is " сожалению" which means "it is a pity" or "it is a shame" as you suggest or, what is my preference "unfortunately," but it does not suggest an apology or request for forgiveness. Whereas Извини(те) or Прости(те) are synonyms that mean I am sorry and may request forgiveness or suggest an apology, as in "I am sorry" or "excuse me." So for example, if you are telling a story you might say сожалению, aвтобус опоздал, поэтому я опоздал на встречу. That is, "unfortunately, the bus was late, so I was late for my meeting." But if you bump into someone on the street, you should say Извините or Простите to request forgiveness.
The nuance that is most important in my example is that I used Извините or Простите, not Извини or Прости as in Duolingo. If you don't know someone well enough to use informal address, you have to use the formal conjugations of the verbs. If you don't, you are showing a lack of respect. The default in Duolingo should be the formal versions, not the informal version.
Finally, you ask about Мне жаль. It can be "I am sorry" as in Мне жаль слышать, что ты болен "I am sorry to hear you are sick," more literally, it means "I am sad to hear you are sick. But again, it does not convey are request for forgiveness or apology. It is closer to сожалению than Извините or Простите, but conveys a stronger set of regret than сожалению. You would say Мне жаль if someone died, but сожалению if the bar down the street closed and you can't go there.
One correction to what I said: I forgot to mention that we must precede сожалению with the preposition К; so it should be "К сожалению." I should have added the preposition in my examples above. The noun is сожаление and means pity or regret. But К сожалению means unfortunately. Use of the preposition K, requires the use of the дательный падеж (dative case),
"Жаль" in the meaning you used was not correct although it is translated directly as "Sorry" as well. However "Жаль" means that someone is sorry for something that happened and he/she is feeling sorry for such to had happen, or said so simply being polite: eg. "I heard you missed your train! I am sorry to hear that" - "Я слышал, что ты опоздал на свой поезд! [Мне] Жаль [слышать это]"
You have various choices, but all require some setup, i.e., installation of files. The easiest is the Russian keyboard available in Windows 10. I do not like it, however. Since I am a touch typist in English, I prefer a keyboard that is phonetically as in English. This means, for example, if you want to type the Russian word for restaurant, the first letter you would type on your keyboard is the R letter--it will appear as a р since the letter p in Russian sounds like an R in English. If you will do a lot of typing in Russian, however, it may be better to learn to touch type in Russian and install the standard Russian keyboard.
Based on the recommendations of my former Russian teachers, I have used one of the keyboard choices available from the website of Paul Gorodyansky at the link: http://winrus.com/mod_e.htm. He provides detailed instructions and both standard and phonetic Russian keyboards are available as choices.
Once the Russian keyboard is installed, you can tab back and forth between English and Russian keyboards by hitting the Windows key and the Space bar simultaneously.