I'm not sure what the OP is trying to gain, as I already answered the Tevilat Yadayim aspect, in my other answer.
To answer his specific queries:
A) Assuming the water comes from a reservoir, is a reservoir a kosher mikveh?
There are basically ways to make a mikveh:
Either a pool of rain-water that is not flowing, or pool of Natural-Stream water that is still flowing and connected to its source.
I don't see how a reservoir meets either; it's mainly river water collected through pipes - and for pipes see below.
But even if it's rain water, it arrived via pipes, and it's flowing (to your tap).
B) Is there a constant connection of water from the reservoir to the (cold water) tap, or is it broken at some point along the way?
It's probably connected, otherwise you wouldn't get good pressure. However, if there's anything like a water pressure booster in the system, it probably breaks the halachic connection.
Also, are there any other potential problems with this solution?
Yes. Here are some possibilities:
1. Pipes
Pipes are possibly vessels, and that would make your Mikveh-extension passul.
See Yoreh De'ah סימן רא - דיני המקוה ומימיו.
לו: צִנּוֹר שֶׁאֵין לוֹ ד' שָׂפוֹת אֵינוֹ חָשׁוּב כְּלִי וְרָאוּי לְהָבִיא עַל יָדוֹ מַיִם לְמִקְוֶה; וְאִם חָקַק בּוֹ גּוּמָא אַחַת קְטַנָּה קֹדֶם שֶׁקְּבָעוֹ, אִם הוּא שֶׁל עֵץ אֲפִלּוּ אֵין הַגּוּמָא מַחֲזֶקֶת, אֶלָּא כָּל שֶׁהוּא נַעֲשֶׂה כֻּלָּהּ עַל יָדָהּ כְּלִי וְכָל הַמַּיִם שֶׁעוֹבְרִין עָלָיו חֲשׁוּבִין שְׁאוּבִים; וְאִם הוּא שֶׁל חֶרֶס, אֵין הַחֲקִיקָה פּוֹסַלְתּוֹ אֶלָּא אִם כֵּן הִיא מַחֲזֶקֶת רְבִיעִית, וְאִם נָפְלוּ צְרוֹרוֹת אוֹ עָפָר בַּגּוּמָא, אֵינוֹ חָשׁוּב סְתִימָה לְבַטְּלוֹ מִתּוֹרַת קַבָּלָה, אֶלָּא אִם כֵּן יִהְיוּ מְהֻדָּקִים לְתוֹכָהּ. סִלּוֹן שֶׁהוּא צַר מִכָּאן וּמִכָּאן וְרָחָב בָּאֶמְצַע, אֵינוֹ חָשׁוּב קַבָּלָה לִפְסֹל בּוֹ הַמִּקְוֶה.
- The pipe cannot be closed on all 4 sides - a classic pipe has 2 openings.
- A U-shaped bend in the plumbing may create a 4 walled pipe and invalidate your connection.
- If the pipe has a curvature in it to collect water, it invalidates your connection.
- A pipe that goes wider and then narrow again is OK.
2. Your sink is not a keli
Your sink - with the drainage hole in the bottom - is probably not a Keli - even with the plug in place.
See, for example, Yoreh De'ah ibid, that once there's a hole in the bottom of a vessel, it's not a vessel, even if it causes a leniency.:
מ: כְּלִי שֶׁנִּקַּב בְּשׁוּלָיו, אֲפִלּוּ כָּל שֶׁהוּא, אֵינוֹ חָשׁוּב כְּלִי לִפְסֹל הַמִּקְוֶה.
See the rest of the long section מ that if it's got a hole - even if it's plugged up (unless it's a permanent plug) - that it's not a vessel.
וְאִם עֵרַב סִיד וּצְרוֹרוֹת וְסָתַם בָּהֶם הַנֶּקֶב, לָא חֲשִיב סְתִימָה לְהַחֲזִירוֹ לְתוֹרַת כְּלִי. אוֹ אִם הוֹשִׁיבוֹ עַל גַּבֵּי הָאָרֶץ, וַאֲפִלּוּ עַל גַּבֵּי סִיד וְגִפְסִיס, לָא חֲשִיב סְתִימָה
3. Water falling vertically
Your connection to the Mikveh (reservoir) is with running water - and we have the rule that קטפרס אינו חיבור - water falling vertically isn't considered connected.
Look in Orach Chaim סימן קנט - באיזה כלי נוטלין הידים, וכיצד יבואו המים לידיו at the end of ז that it's a dispute as to whether we consider a vertical drip of water a connection for washing hands.
דְּנִצּוֹק זֶה מְחַבֵּר הַכְּלִי לַיְּאוֹר וַחֲשִׁיבֵי מַיִם שֶׁבַּצִּנּוֹר מְחֻבָּרִים לְמֵימֵי הַיְאוֹר, אַף עַל גַּב דְּנִצּוֹק כָּזֶה לֹא חָשִׁיב חִבּוּר לְעִנְיַן טְבִילָה. וְיֵשׁ חוֹלְקִין בָּזֶה וְאוֹמְרִים דְּגַם לִנְטִילַת יָדַיִם לֹא חָשִׁיב חִבּוּר
According to the second opinion, your sink is not connected to the reservoir.