Mr. Raghavendra,
You seem to have opened a very interesting thread. While Ms. Meenakshi vouches that Partnership firm is not a legal entity, others support me at least on the point that a firm is entitled to make an application. It is true a P FIrm is not a distinct legal entity apart from the partners constituting it. But in the context of holding a trademark in the name of a firm, the law permits it.
Coming to your further queries, a prior user will have a better right. If you have adopted a trademark and put it to use at an earlier point of time, your rights are protected. In Corn Products case ( AIR1960 Supreme Court 142) the Supreme Court said “the presence of a mark in the register does not prove its user at all. It is possible that the mark may have been registered but not used. It is not permissible to draw any inference as to their user from the presence of the marks on the register” It therefore follows that a prior user will havea better right than a prior registered owner not using the mark.
So far as your doubt on web name is concerned, you have to register your domain irrespective of the fact that you have a registered trademark. In other words, on the ground that you are a registered trademark owner, the domain name would not be kept reserved for you. In your words, it is not ( necessarily) obvious that your are the only one who can have a domain in your name.
You may please note that if you are a registered owner of a trademark and someone have registered a domain comprising your trademark without bona fides, you can complain to ICANN - Internet Corporation for Assigned Names and Numbers and get the same transferred in your name if you can prove better title for the domain in your favour.
If some other firm dealing in similar products/services launches its products/services with a exactly the same name, you can file a case for infringement of trademark if the mark is regisgtered in your favour. In a case of very identical name, you can file a PASSING OFF suit on the basis of a prior adopter and user of the mark and show that the public tend to confuse between your mark and the offending mark.