There's a lot of different boiler designs...it sounds like yours has fairly high mass and isn't designed for cold starts (i.e., it maintains a set heat band whenever on). The newest, low-mass boilers are generally designed to shut down when there's no call for heat. Mine, for example, in the summer often only runs once a day after the morning showers, then cools off and often reaches ambient temps before it needs to turn on again.
There's a tradeoff between response, longevity, and reliability for the settings. Many older boilers cannot handle low, condensing, return water temps. So, to keep them from self-destructing, you have to maintain some minimum temperature, often in the 140-degree range or so. You could reduce the low-temp setting during the non-heating season, and may also be able to reduce the upper bound. Most gas-fired WH are not all that efficient - your boiler is generally better. You are taking a hit if it is trying to maintain heating season temperatures during the summer, though.
To maintain the same hot water supply, you'd need a bigger tank if you go direct fired...indirects take advantage of the entire boiler's heat capacity and can recover much quicker so they generally are smaller. You may find that alone a consideration for your small space. Also, you have to consider the additional flue - you may not be able to connect the new WH's flue into the boiler's. Indirects come in at least a couple of different designs: jacketed and coils. Either can develop leaks. There are some, though, that have very long warranties. Mine is a SuperStor Ultra, mostly SS, and I expect it to last a very long time.
One of the pros will be familiar with that model and be best able to offer some practical suggestions, but my gut feeling is that another indirect may be the better long-term choice, especially if you eventually consider a new, more efficient, low-mass boiler in the future as well.