1. I wonder if that's some kind of feature that your carrier provides, as Golfdriver97 mentioned above. I have Verizon, and don't see that option anywhere, either on my phone or on the Verizon website under my account. Try logging into to your account on your wireless provider's website, and search the texting features to see if there is some option for text messages to be sent to your email as well.
2. On my Facebook app, I find that I can't directly download photos either--long pressing doesn't seem to do anything. If you bring up a photo that someone else posted, then tap the menu button, you'll get a few options including "Share" (which only shares it on your Timeline) and "Set as," which then gives you the option of making that picture a wallpaper or a contact picture. I haven't tried, but I suppose that if you set it as a wallpaper, you could somehow retrieve that photo and place it in your gallery as well, but not sure if the photo would end up being stored in some weird location. Interestingly, Google+ also has this limitation.
3. I'm assuming you only have voicemail that you can access by dialing the phone. For most systems, you do have to dial in, then listen and then delete the voicemail. You can always fastforward to the end (for Verizon, dial "33") then delete (dial "7"), so you don't have to listen to the whole message. Verizon has an additional paid feature called Visual Voicemail, which transcribes your voicemails that you can then read in an app--I think you can then tap to delete those messages, and your actual voicemail will delete as well. But you have to pay extra, so it isn't worth it to me. If you're on a different wireless service, see if they have something similar. For example, my Comcast cable phone service allows me to access voicemails on the web and on the Xfinity Connect app, where I can listen and delete very easily by tapping.
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