And Also another question, It looks It has been cleaned too, can a coin have Lost some Weight from cleaning and can cleaning like this remove details from the coin?
Minor differences in weight happen from time to time. In my opinion the weight is not off by enough to throw up any red flags.
There are some really good fakes out there. But this one seems legit. The weight most likely is light due to harsh cleaning. I have a couple denarii in the same condition and weight range.
If you search on ACSearch by including only the first two digits of the weight, e.g., 2.7, it will show results for any coins described as 2.70-2.79. There are 493 results in that weight range (including some duplicates and inappropriate returns). There are 298 results in the 2.60-2.69 range. So, your coin is not exceptionally underweight, but it is down on the low end relative to coins appearing in single lots at major auctions (probably in the lightest 2-4% of all Trajan Denarius results, though you could refine that result in various ways). Incidentally, you can use that method to construct a rough frequency distribution of weights for any kind of coin with enough recorded cases in the database. (E.g., 2.80-2.89 = 838 results... the peak, as you suggested, is between 3.10 and 3.39, with each 0.10g interval returning 3,000 - 3,100 results.) Or to assist in provenance searching. (First thing I do when researching any coin is search for other specimens withing a few 0.01g on either side to see if there are any sales in ACSearch that I don't know about.)
It also depends on the cleaning process used. Some people still use harsh, chemical cleaning processes on batches of silver to strip off any "gunk" quickly. This strips off everything, including patina but can have the problem of removing copper from the alloy, losing weight in the process.
This is the most extreme examples I have owned which I will share as illustration. This denarius (two different images of the same coin under different setups) of Septimius Severus is extremely light (1.459g) but superficially looks like good silver. The coin is clearly struck and not a cast fake. The surfaces have a "frosted" look that I now link with harsh chemical cleaning. The only real evidence of the loss of copper from the alloy of the coin is the light weight. You would have to perform a metallurgical analysis on the alloy of the coin to verify this but for such an example it isn't worth the effort or expense.