Is the panchanga the same everywhere?

The sequence is, but the times are not. Tithi and nakshatra boundaries fall at particular moments in universal time, so the local clock time differs by longitude and timezone, and the tithi in effect at sunrise can differ between cities.

What is the difference between nakshatra and rashi?

A rashi is one of twelve 30-degree signs. A nakshatra is one of twenty-seven 13°20' lunar mansions. They divide the same circle differently and do not align, so a nakshatra can straddle two signs.

Panchanga

The panchanga is the Vedic almanac, and its name means five limbs. Each limb is a different way of measuring the relationship between the Sun and the Moon on a given day, and together they describe the quality of that day.

The five

Tithi — the lunar day, one of thirty, measured as each 12 degrees of separation between Moon and Sun. Vara — the weekday, ruled by its planet. Nakshatra — which of the twenty-seven lunar mansions the Moon occupies, each 13°20' wide. Yoga — one of twenty-seven, from the sum of the Sun's and Moon's longitudes. Karana — half a tithi, eleven of them cycling through the month.

Why the boundaries move

None of these limbs aligns with the civil day. A tithi is complete when the Moon has gained another 12 degrees on the Sun, and because the Moon's speed varies, a tithi can run from about 20 to about 26 hours. So a tithi begins and ends at a clock time that differs each day and each place. The same is true of the nakshatra, which is why an almanac gives times of day rather than simply naming the day's nakshatra.

The nakshatra a chart depends on

Of the five, the nakshatra matters most for a birth chart: the Moon's nakshatra sets the entire Vimshottari dasha sequence, and its lord becomes the first mahadasha ruler. It is also the basis of most of the Guna Milan score. A birth near a nakshatra boundary makes both of those sensitive to the accuracy of the recorded time.

Questions

Is the panchanga the same everywhere?: The sequence is, but the times are not. Tithi and nakshatra boundaries fall at particular moments in universal time, so the local clock time differs by longitude and timezone, and the tithi in effect at sunrise can differ between cities.
What is the difference between nakshatra and rashi?: A rashi is one of twelve 30-degree signs. A nakshatra is one of twenty-seven 13°20' lunar mansions. They divide the same circle differently and do not align, so a nakshatra can straddle two signs.

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Vimshottari dasha — How the mahadasha sequence is calculated from the Moon's nakshatra, the nine planetary periods and their lengths, and how antardasha subdivisions are derived.
Kundali matching (Guna Milan) — The eight kootas of ashtakoot matching and their 36 points, what Bhakoot and Nadi actually test, and how Mangal dosha is assessed.
The kundali (birth chart) — A janam kundali is a map of the sky at a birth moment. How the rashi chart is built, why birth time matters to the minute, and how sidereal positions are computed.