Conversion to Islam

There are all sorts of reasons why people convert to a religion, and sometimes the person doing the converting doesn't even know all of their own psychological motivations. 

If he professes and accepts the shahadatayn (there is no god but Allah and Muhammad is his prophet) and the general beliefs of Islam that go with it, and appears to be sincere and honest about it, then he is considered a Muslim and she is allowed to marry him. Only Allah knows what it is in people's hearts, and it sounds like this guy may be rather complicated. 

If he is openly telling her that the only reason he is practicing Islam is because he wants to marry her (and, for instance, he will apostasize if she rejects him), then it is inadvisable for her to marry him, especially if she intends to have Islam as part of the family environment and pass Islam on to the children. For instance, if he says he doesn't really believe that the Qur'an is divine revelation but is just saying it to her dad to marry her, that's more or less like saying he is not a Muslim anyway. 

However I suspect that, after 7 years, the truth here is probably somewhere in the middle - that is, if there were no commitment or self-investment in the faith, he probably wouldn't still be practising it. From the little you have said, it sounds like his personal identity and sense of self may be caught up in her, such that he defines himself by her, including but not limited to religious identity. (As a side note, this isn't unheard of for converts to Islam, given that the often lose a sense of family or heritage, although in my experience it tends to happen more with women.) That doesn't make him a non-believer, but it isn't psychologically healthy especially in a marriage. (This may or may not be the case, just putting it out there.)

In any case, it is good for him to sort himself out. It may be that Allah will test him by preventing the marriage in order to see whether he is really dedicated to his faith. If he wouldn't continue the faith without her, he should question why he wants to commit to starting a Muslim family and living Islam for the rest of his life and having it live on in his children.

7 years is a rather long time to follow a religion solely for the sake of hoping for a marriage and, indeed, it is a rather long time to wait for someone. Yes it is somewhat romantic to think of someone waiting that long, but, also, when that happens, sometimes it is the case that the person is more in love with what they can't have rather than the actual person, or else they are in love with the ideal of the person instead of the actual person. It may be than an actual marriage wouldn't work out, even if it is what he thinks he wants. Also, if there has been no progress towards the marriage in 7 years, it is worth considering that it may never work out.

So I think it is better if he focuses on some self-reflection on deciding who he really is or believes and who he himself wants to be in life, rather than putting things on the shoulders of his beloved. It's not really fair to put all that on someone else anyway. Regardless of what the interfering issues with the marriage are (or what he thinks the interfering issues are, since, surprisingly often, in my experience, men do not understand the real reasons why women do not want to marry them), I also think he would benefit from setting a deadline for the marriage, say, 6 months, either they get married or he moves on.

According to Islam, it is not ideal to have a pet dog, and it is better in terms of Islamic etiquette not to have a dog in the house, but it is not forbidden to have one.

However, if you have a dog in the house, you should be sure to observe Islamic rules of cleanliness for prayer. For instance, before praying salat, wash any area of yourself that the dog might have been licking or if the dog came into contact you while it was wet. When praying salat, avoid praying directly on the floor and instead pray on top of something (like a prayer rug or, if you do not have one, a towel or mat or something similar), and put it away when you are not using it so the dog cannot get to it. Also, be sure your clothes do not have dog hair on them and have not been licked by the dog or touched by a wet dog; if your dog is very affectionate, you may wish to have a separate pair of clothes that you only wear for salat (if this is possible). Also, if your family is very casual about dishes and where the dog goes, be sure the dishes you use have not been, ahem, slobbered on, licked, etc. (Some people are very strict about these things and some people are casual, that is why I am mentioning it)

Yes, this is inconvenient, which is one of the reasons it is better not to have a dog in the house! But you can still practice Islam and all aspects of Islam with or without a dog present.