can we refer to EMV & Chip Card as the same thing?
For practical purposes, you can if it's your industry. I've found that retailers undergoing a shift in payment technology may refer to them as EMV cards, or chip cards, or smart cards, regardless of whether or not the terminology is correct.
What are the problems that having a chipcard or an EMV standard complianced card solve?
Compliance with the EMV standard means that different chips made by different foundries, cards created by different manufacturers, and cards issued by different banks can all interact with many different payment terminals made by different companies and installed at different retailers. EMV is the protocol that binds them all together.
An EMV transaction is a defined flow of information between the retailer, the retailer's terminal, the card, the cardholder, and the bank. The cardholder inserts the card in the terminal. The retailer's terminal exchanges data with the card. The terminal prompts the user to enter their PIN, and passes the PIN into the card; the card exchanges signed and/or encrypted data with the bank, and ultimately the card provides a digitally signed authorization to the retailer that says "yes, I approve this transaction for $100". The trick is that the card contains a secret key, put there by the bank, and that key is inaccessible to anyone else. Only that secret key plus the secret PIN associated with that key can properly generate the needed valid digital signatures on the approval data. The retailer won't accept the transaction unless it's properly signed.
This entire interaction is governed by the EMV standards, and is already implemented in millions of POS terminals and billions of smart cards. The EMV standards enable more manufacturers, more banks, more retailers, and more customers all to exchange payments for goods and services.
How do they solve the problems of other technologies such as proximity cards (Can't proximity cards have processors embed with them? )
Many cards have processors embedded in them, and there are many wired and wireless technologies for communicating with these smart cards. Some of these other technologies have some superior features -- RF cards are immune to dust, for example -- and some of them have drawbacks -- NFC cards can be read through your pocket and clothing on a bus or subway, and all wireless cards have extra circuitry that the EMV cards don't need. Many of those technologies weren't present when EMV was first defined, so EMV may not be able to work with all of them yet. Some of those other technologies may have other failings that make them undesirable for EMV.
EMV contact cards are among the more secure technologies, especially given the many limitations on smart cards: they have to be super cheap to distribute to thousands of customers, the readers have to be fairly affordable, they have to be extremely secure, they have to work in many different environments, they have to be reliable, and they still have to be super cheap. It's not that other technologies can't or won't work; it's that EMV is an international standard that's been in use for over a decade, and is already installed in millions of retail establishments around the globe.
what about magnetic strip cards (least of my worries cus it's been answered by other posts!)?
Mag stripe cards have never been secure. They are static: they always return the same data, read after read. You can copy the data from them and reuse it. A smart card, on the other hand, generates a dynamic digital signature every time you use it, assuring you that this is a card that knows the secret. And that secret is never revealed outside of the card environment. This makes them very difficult to clone. Plus, an attack on one card is slow and hard, and it teaches you nothing about the secrets inside the next card. That means attacking one card does not give you any information about attacking the next card. But if you are in a place where you can attack and copy one mag stripe card, copying 100 is just as easy.