On the whole, drug resistant bacteria will likely remain as capable in other characteristics. Due to the complex nature of DNA (specifically gene functions), genetic codes tends to be multipurpose. When select for any one attribute (e.g drug resistance) you could potentially affect zero or more other attributes. You just won't know which ones (based on today's bio-genetics understanding). So there is no reason to expect drug resistant bacteria to be weaker.

You are right bacteria have proven very cable at appropriating useful DNA from across the species. Also, we can reasonably expect drug resistant bacteria to spread just like other bacteria spreads (i.e varied methods and levels of difficulty). Among the factors that affect the spread of bacteria is the mobility of the host. If a host dies in timely fashion the bacteria spreads less and if they linger around longer the opportunity to spread increases. Like other infectious pathogens, this is mainly species dependent. When a multitude of antibiotic resistant bacteria spread around sufficiently are a common stay in the ether then they become part of the random bacteria that would likely infect if you just got a cut.