Dear Sir or Madam, 

I am going to go to Seville in the first weekend of December and I don’t have any place to stay there. I am writing to you because I would like to know information about your hotel. (We always capitalize the pronoun “I." Also, it sounds better to say: "I don't have a place to stay" instead of "I don't have a place to sleep," even though technically you are indeed looking for a place to sleep).

Firstly, I am interested in knowing if there will be rooms available on these dates and if they are avaliable to me. (We usually use “firstly” when making a point. “Firstly, what you’re suggesting is wrong…” But it’s usually safer to stick with “first." It's possible the Brits use "firstly" more than we do in America, I don't know).

Secondly, I wonder if your hotel has any activities for children under four because we are a family with a son and we will have some free time. (The traditional rule with numbers is that we spell out numbers one through ten and use Arabic numerals for 11 and above).

Thirdly, would you also mind telling me what places we should visit there? Furthermore, we are very interested in the food and the amenities of the hotel, because my wife loves spas and we would like to have a spa treatment. ("Installations" in America are usually pieces of equiptment, or artwork; "amenities" are the features or facilities that belong to a place of business).

Finally, I hope you might also let me know about the price of the whole weekend. Thanking you in advance for your assistance. ("Prize" is something you win; "price" is what you pay for something). 

I look forward to your reply at your earliest convenience.

Faithfully, (Try "sincerely, sincerely yours, regards, or kind regards" instead. "Faithfully" sounds a little romantic.)

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¡Tú inglés es mucho más mejor que mi español, y esto es my bueno, Mario! :-)

Formal letter

Dear Sir or Madam,

I am going to go to Seville in the first weekend of December and I have not booked any accomodation yet. (Its more natural like this) I am writing to you because I would like to know more information about your hotel.

Firstly, I am would like in know if there will be available rooms available for this date. and if there are offers to me.

Secondly, please could you tell me if the hotel has any activities for children under 4 years old? because we are a family with a young son. and we would want to have some free time.

Thirdly, would you also mind telling me what places we could visit there? Furthermore, we are very interested in the food and the facilities of the hotel, because my wife loves to go to the spa and we would like to have a spa treatment.

Finally, please could let me know about the price of the whole weekend. Thanking you in advance for your assistance.

I look forward to your reply at your earliest convenience.

Yours faithfully

Blue = suggestions Red = corrections.

Well written you only have some small mistakes.

Formal letter

Dear Sir or Madam,

I am going to go to Seville in the first weekend of December and I don’t have any place to sleep there. I am writing to you because i would like to know information (either "know more" or "find out information")about the hotel.

Firstly, I am interested in knowing if there will be available rooms for this date and if there are offers to me any offers available. (No need to say "to me," that's understood.)

Secondly, I wonder if this hotel has any activities for children under 4 years old because we are a family with a son and we would want to have some free time.

Thirdly, would you also mind telling me what places we should visit there? Furthermore, we are very interested in the food and the installations of the hotel, because my wife loves the spa and we would like to fun find a spa treatment.

Finally, I hope you might also let me know about the prize price of the whole weekend. Thanking you in advance for your assistance.

I look forward to your reply at your earliest convenience.

Faithfully,

Very well written!  This is just a personal preference, but even in a formal letter, I would use first, second, third, etc. instead of firstly, secondly, thirdly.  Both are correct, but at least in American English, the -ly is pretty uncommon.  Firstly is especially rare.  Some start with first, then use secondly, thirdly, etc.    

Formal letter

Dear Sir or Madam,

I am going to be in Seville for the first weekend of December, and I would like to reserve a room for my wife, myself and my 3 year old son for the Friday, Saturday and Sunday nights, leaving on the morning of Monday, the 8th of December.

Do you have anything available for those 3 nights, and could you inform me about any special offers that might be available over that time?

Also, could you tell me if there are any things available in the hotel that might interest my son, such as a pool or an activity centre. And do you have any spa treatments or something like that that would interest my wife?

(They are unlikely to take the time out to tell you about the food in the restaurants in the hotel or how to get to the Alcázar Gardens. Actually, now that I think about it, don't people just phone the hotel to make reservations nowadays; I don't think it is likely that they will write you a letter back. Actually, now that I really think about it I think that everyone makes their reservations on line!)

(The price will be stated when they reply to the bit about the special offers.)

Thank you for your assistance.

I look forward to your reply. (If they don't do it at their earliest convenience i.e. now they will just forget to do it completely.

Yours faithfully,