Hello David.
I am Ukrainian and I've recently moved to South Africa to my boyfriend who had the same concerns as you have now. Also, I've worked in marriage agency in Ukraine and I can quite understand whether woman is serious and sincere or she just uses a man.
If this woman lives in a small town, there might be a problem with the Internet, this is true. But the middle monthly fee is about 80-100 UAH. If she goes to some Internet cafe, the price per hour can be about 5-10 UAH. I am not sure, as I used Internet at home.
Sending some stuff like phone is not a good idea at all, because it is very common that things get stolen while the delivery, the sad truth.
Bank account is not as widespread thing in Ukraine as abroad. Many people get their salaries straight into hands from the accountant.
I met my boyfriend through work (my company provided software for the company of my boyfriend), we started to chat, then talked through skype and then he decided to come. After 6 months since our acquaintance, I've moved to SA and now we live together, absolutely happy. So, believe me, that there are many wonderful women in Ukraine. But still there are woman who use foreigners to have a better life. So, be attentive and listen to your heart and mind.
Hope this helps.
Let me know if you have any questions 
Anna