Be direct about what you want, why you want it, and what play experience you imagine for your character (or the party as a whole)
Those are in increasing order of importance for me as a DM. The mechanics of downtime can be odd to work with, especially when time is a fluid resource. Working downtime into a game can be a lot of work for a DM. Before putting in that effort I like to be sure that my players actually want the downtime, and have some guidance into how I should prepare and manage it to make a fun experience for them.
Some players just want pure RP opportunities while gambling around town to immerse themselves in the setting, while others want to run businesses based around their characters' abilities, and still others want detailed downtime schedules that fit with supplemental rules like those detailed in Strongholds and Followers.
Those are the angles I suggest using to ask your DM for downtime: downtime is fun for you, and you were hoping to have opportunities to roleplay your character in that way as well as the content that she's already offering.
I would definitely not assume that it will be to your DM's benefit, nor to the entire group's benefit, to include downtime like you are seeking. It might be true, especially if your entire group would enjoy the downtime elements of the game and your DM has the time and interest in including them. But if you haven't verified that then it might be a difficult tack to suggest that your DM will benefit from doing lots of additional work that her current adventure resources don't necessarily support very much (if at all).
It's also worth emphasizing elements of the game that you have been enjoying, as some DMs can receive this sort of request as a criticism (whether or not that is the intent). Asking for an additional element to be included in a game is a request; telling a DM that they are making a mistake or doing a bad job because they have neglected a fairly tangential, optional element of the game which you want can be unpleasant, if not condescending, and is not good form for a player-DM relationship.
I've been having fun playing my character through this adventure! But I built my character with the idea that they'd be able to engage in some downtime activities, like [your specific examples]. I don't know what plans you have for the rest of the campaign, but do you think that we might have some opportunities to do that sort of thing? I really like the idea that my character can [spend downtime doing X, in order to enable outcome Y, which satisfies my gameplay goals like Z].
Downtime activities can be awkward, confusing, and unsatisfying for players in 5e
I've been on both sides of this issue, as a DM and as a player, and my observations align in each case. The biggest issues I've noticed are that:
- Downtime is player-driven. Players decide on their own goals and
methods, which can also lead to feelings of aimlessness and a feeling
that the game lacks direction. This can be problematic for many
reasons but, importantly, can detract from some of the most important
responsibilities the DM has (like to keep the game moving). Players that are excited about downtime activities will be fine, but those activities can be very disruptive and dull for players that are not as interested
- Downtime can easily be unsatisfying, mechanically. It can be
narratively interesting, but outside of roleplaying it often returns
money (which players have limited meaningful opportunities to spend)
or items and situations which the game doesn't simulate very precisely. In
all cases, it tends to require extra effort on the DM's part to
include and it's brutal to spend a great deal of time and effort to include a system that players grow to hate and/or ignore
- If efforts are made to ensure that the products of downtime efforts
are meaningful to the game in some way, downtime becomes another
resource for the DM to track and manage (too much can unbalance
things in ways difficult to predict or repair). Published guidelines
offer varying help with doing this, but things often fall to homebrew
at some point. It is not necessarily a trivial effort for the DM to
do all of this
None of this means that downtime activity is bad (it isn't!), but they are real reasons that a DM might want to, and even be wise to, be cautious about including it as a matter of course. 5e is not 3.5e, in which characters could essentially have day jobs (or side jobs) practicing a profession for specific, mechanically-defined benefits between adventures or while journeying.
These reasons are why I would be very hesitant to suggest that ample opportunity for downtime activities are things that a DM definitely must or should include in all games for all players. When asking a DM to include new dimensions of gameplay it is worth presenting them with information on what you want and why so that they can be confident that they are actually delivering that to you. Especially if there may be substantial effort involved in designing and delivering it.